New Recursive Representations for the Favard Constants with Application to Multiple Singular...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 523618 12 pageshttpdxdoiorg1011552013523618

Research ArticleNew Recursive Representations for the FavardConstants with Application to Multiple Singular Integrals andSummation of Series

Snezhana Georgieva Gocheva-Ilieva and Ivan Hristov Feschiev

Faculty of Mathematics and Computer Science Paisii Hilendarski University of Plovdiv 24 Tzar Assen Street 4000 Plovdiv Bulgaria

Correspondence should be addressed to Snezhana Georgieva Gocheva-Ilieva snegochevayahoocom

Received 11 February 2013 Accepted 22 April 2013

Academic Editor Josip E Pecaric

Copyright copy 2013 S G Gocheva-Ilieva and I H Feschiev This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

There are obtained integral form and recurrence representations for some Fourier series and connected with themFavard constantsThe method is based on preliminary integration of Fourier series which permits to establish general recursion formulas for Favardconstants This gives the opportunity for effective summation of infinite series and calculation of some classes of multiple singularintegrals by the Favard constants

1 Introduction

The Fourier series and related with them Achieser-Krein-Favard constants often simply called Favard constants havesignificant theoretical and practical roles in many areas [12] These remarkable mathematical constants are introducedfirstly in the theory of Fourier series and approximations offunctions by trigonometric polynomials [3]

The classical definitions of Favard constants are given bythe infinite series [1 4 5]

119870119903=4

120587

infin

sum

]=0

(minus1)](119903+1)

(2] + 1)119903+1 119903 = 0 1 2 (1)

119903=4

120587

infin

sum

]=0

(minus1)]119903

(2] + 1)119903+1 119903 = 1 2 (2)

These constants find wide applications in the approxima-tion theory for exact and asymptotic results on the approx-imation of functions and especially for the best approxi-mations of trigonometric and other classes of functions indifferent spaces and related inequalities [1 5ndash14] In partic-ular many important applications are concerned with theapproximation of Euler cardinal periodic and other type of

splines [15ndash17] It can be noted that Favard constants are con-nected to approximations that are best in a pointwise sensein comparison for instance with the Lebesgue constantswhich are connected to approximations that are best in aleast-squares sense (Fourier series) [5] The Favard constantsplay also an important role in estimating optimal quadratureand cubature formulas calculation of singular integralssome classes of differential integrodifferential and integralequations [18ndash24] and in other areas

Nevertheless widely used as a whole the properties of theFavard constants have not been investigated well enough [14]except for some particular cases

Different methods for their calculation are given forinstance in [2 Ch 52] In general thesemethods are based onthe properties of the well-known constants and special func-tions as gamma function Γ(119911) generalized Riemann zetafunction 120577(119911 119886) the Bernoulli polynomials 119861

119899(119909) and the

Bernoulli numbers 119861119899 and the Euler polynomials 119864

119899(119909) and

the Euler numbers 119864119899 given by the following expressions [2]

Γ (119911) = int

infin

0

119905119911minus1

119890minus119905

119889119905 (Re 119911 gt 0)

Ψ (119911) =Γ1015840

(119911)

Γ (119911)

2 Abstract and Applied Analysis

120577 (119911 119886) =

infin

sum

119896=0

1

(119896 + 119886)119911=

1

Γ (119911)int

infin

0

119905119911minus1

119890minus119886119905

1 minus 119890minus119905119889119905 (Re 119911 gt 1)

119861119899(119909)

119905119890119909119905

119890119905 minus 1=

infin

sum

119896=0

119861119896(119909)

119905119896

119896(|119905| lt 2120587)

119861119899= 119861119899(0) 119861

119899(119909) =

119899

sum

119896=0

(119899

119896)119861119896119909119899minus119896

119864119899(119909)

2119890119909119905

119890119905 + 1=

infin

sum

119896=0

119864119896(119909)

119905119896

119896

(1

cosh (119905)=

infin

sum

119896=0

119864119896

119905119896

119896) (|119905| lt 120587)

(3)

From now on we will use the following notations

119860119899(119909) = int

119909

0

119889119909 sdot sdot sdot int

119909

0

ln tan 1205792119889120579 (0 le 119909 le 120587)

119861119899(119909) = int

119909

0


119909

0

ln(2 sin 1205792) 119889120579 (0 le 119909 le 2120587)

119862119899(119909) = int

119909

0


119909

0

ln(2 cos 1205792) 119889120579

(minus120587 le 119909 le 120587) (119899 = 1 2 3 )

(4)

119863119899(119909) =

119909119899

119899 (5)

119879119903=

infin

sum

119899=1

1

119899119903(119903 = 2 3 )

119876119903=

infin

sum

119899=1

(minus1)119899

119899119903(119903 = 1 2 3 )

(6)

where the threemultiple singular integrals (4) contain exactly119899 integral operations

The main purpose of this paper is to establish newrecursive formulas for the Favard constants (1) including onlyfinite number of terms and recursive formulas for (2) Theywill be further used to obtain new integral representations forthe previously stated objects in particular for calculation ofthe multiple singular integrals (4) and summation of series

2 Recursive Representations for119870119903

and Some Fourier Series

We will prove the following

Theorem 1 For the constants119870119903(119903 = 1 2 3 ) the following

recursive representations hold

1198702119904+1

=1

2[(minus1)

119904

1198632119904+1

(120587) +

119904

sum

119901=1

1198702119904+1minus2119901

(minus1)119901minus1

1205872119901

(2119901)]

(1198700= 1 119870

1=120587

2)

(7a)

1198702119904= (minus1)

119904

1198632119904(120587

2) +

119904

sum

119901=1

1198702119904+1minus2119901


(2119901 minus 1)(120587

2)

2119901minus1

(119904 = 1 2 3 )

(7b)

Proof We will use the method of induction and preliminaryintegration of appropriate Fourier series Let us start by thewell-known expansion [2]

4

120587

infin

sum

]=0

sin [(2] + 1) 119909]2] + 1

= 1 (0 lt 119909 lt 120587) (8)

where for 119909 = 1205872we will have1198700= 1 By integration of both

sides of (8) in [0 119909] we get

minus4

120587

infin

sum

]=0

cos [(2] + 1) 119909](2] + 1)2

+4

120587

infin

sum

]=0

1

(2] + 1)2= 119909 = 119863

1(119909)

(0 le 119909 le 120587)

(9)

For 119909 = 120587 (9) gives us 21198701= 120587 = 119863

1(120587) and conse-

quently1198701= 1198631(120587)2 = 1205872 The same results for119870

0and119870

1

can be achieved starting from the equality [2]

infin

sum

119899=1

sin 119899119909119899

=120587

2minus119909

2=120587

2minus1

21198631(119909) (0 lt 119909 lt 2120587)

(10)

For 119909 = 1205872 we find again that 1198700= 1 After integration

of the both sides of (10) we have

minus

infin

sum

119899=1

cos 1198991199091198992

+ 1198792= minus

1199092

4+120587119909

2= minus

1

21198632(119909) +

120587

21198631(119909)

(0 le 119909 le 2120587)

(11)

For 119909 = 120587 we find that 1198701= 1205872 Next after integration

of the both sides of (11) we obtain

minus

infin

sum

119899=1

sin 1198991199091198993

+ 1198792119909 = minus

1

21198633(119909) +

120587

21198632(119909) (0 le 119909 le 2120587)

(12)

which for 119909 = 120587 gives us

1198792= minus

1

21205871198633(120587) +

1

21198632(120587) =

1205872

6 (13)

Abstract and Applied Analysis 3

If we put 119909 = 1205872 in the same equality (12) and make alittle processing we will arrive at the value119870

2= 1205872

8On the other hand after integration of both sides of (9)

we obtain

minus4

120587

infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)3

+ 1198701119909 = 119863

2(119909) (0 le 119909 le 120587)

(14)

Here for 119909 = 1205872 we get minus1198702+ (1205872)119870

1= 1198632(1205872) and

consequently

1198702=120587

21198701minus 1198632(120587

2) =

1205872

8 (15)

For the constant 1198703 we must now integrate both sides of

(12) asinfin

sum

119899=1

cos 1198991199091198994

minus 1198794+ 11987921198632(119909) = minus

1

21198634(119909) +

120587

21198633(119909)

(0 le 119909 le 2120587)

(16)

and put 119909 = 120587 here or integrate both sides of (14) as

4

120587

infin

sum

V=1

cos [(2V + 1) 119909](2V + 1)4

minus4

120587

infin

sum

V=1

1

(2V + 1)4+ 11987011198632(119909) = 119863

3(119909)

(0 le 119909 le 120587)

(17)

and put again 119909 = 120587 Then we will have

1198703=1

1205871198634(120587) minus

3

21198633(120587) +

120587

21198632(120587)

=1205872

41198701minus1

21198633(120587) =

1205873

24

(18)

Going on the indicated procedure on the base of induc-tion we easily arrive at the recursive representations (7a) and(7b) and complete the proof

Remark 2 The scheme of this proof is valid for the most ofthe other statements in this paper

In connection with Theorem 1 we would like to noteanother representation of119870

119903(see eg [25]) It can be written

in terms of Lerch transcendent [25] or as it is shown in [2Section 514]

1198702119904minus1

=2

120587 (2119904)(22119904

minus 1) 1205872119904 10038161003816100381610038161198612119904

1003816100381610038161003816

1198702119904=

2

120587 (2119904)(120587

2)

2119904+1

100381610038161003816100381611986421199041003816100381610038161003816 (119904 = 1 2 3 )

(19)

where the Bernoulli and Euler numbers are specified in (3)Data for values of magnitudes of 119870

119903using (7a) and (7b)

are shown in Table 1The equalities (10)ndash(13) outline a procedure for summing

up the numerical series 1198792119904 (119904 = 1 2 3 ) in (6) It leads to

the assertion

Table 1 Exact and approximate values of the Favard constants 119870119903

calculated by the recursive formulas (7a) and (7b) usingMathemat-ica software package

119903 Exact values of119870119903

Approximate values of119870119903

1 1205872 157079632679489661923132169162 120587

2

8 123370055013616982735431137493 120587

3

24 129192819501249250731151312774 5120587

4

384 126834753950524006818281683185 120587

5

240 127508201993867272192808879186 61120587

6

46080 127267232656453061325614987117 17120587

7

40320 127343712480668316338644619008 277120587

8

2064384 127317548065260581363477696719 31120587

9

725760 1273261242472487546381436665610 50521120587

10

3715891200 1273232382729394849508279710811 691120587

11

159667200 1273241945872154096771507790112 540553120587

12

392398110720 12732387471572495304117396905

Corollary 3 The following recursive representation holds

1198792119904= (minus1)

119904

[1

21205871198632119904+1

(120587) minus1

21198632119904(120587)]

+

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198792119904minus2119901

(119904 = 1 2 3 )

(20)

where for 119904 = 1 by definition 1198790= 0

It can be noted that the numbers 1198792119904can be also repre-

sented by the well known formula (see eg [2 512])

1198792119904=22119904minus1

1205872119904

(2119904)

100381610038161003816100381611986121199041003816100381610038161003816 (119904 = 1 2 3 ) (21)

The same procedure applied on the base of the equalityinfin

sum

119899=1

(minus1)119899sin 119899119909119899

= minus1

21198631(119909) (minus120587 lt 119909 lt 120587) (22)

leads to the following


1198762119904=(minus1)119904

21205871198632119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198762119904minus2119901

(119904 = 1 2 3 )

(23)


In this connection we will note the explicit formula for1198762119904represented by the Bernoulli numbers (see [2 521])

1198762119904=

(1 minus 22119904minus1

) 1205872119904

(2119904)

100381610038161003816100381611986121199041003816100381610038161003816 (119904 = 1 2 3 ) (24)

It is easy to see that the constants119870119903satisfy the following

inequalities (see also [1])

1 = 1198700lt 1198702lt 1198704lt sdot sdot sdot lt

4

120587lt sdot sdot sdot lt 119870

5lt 1198703lt 1198701=120587

2

(25)

and lim119903rarrinfin

119870119903= 4120587


Theprocedure of getting the representations (11) (12) and(16) with the help of (10) gives us an opportunity to lay downthe following

Theorem 5 The following recursive representations hold

infin

sum

119899=1

sin 1198991199091198992119904minus1

= (minus1)119904

[1

21198632119904minus1

(119909) minus120587

21198632119904minus2

(119909)]

+

119904minus1

sum

119901=1

(minus1)119901+1

1198792119904minus2119901

1198632119901minus1

(119909)

infin

sum

119899=1

cos 1198991199091198992119904

= (minus1)119904minus1

[1

21198632119904(119909) minus

120587

21198632119904minus1

(119909)]

+

119904minus1

sum

119901=0

(minus1)119901

1198792119904minus2119901

1198632119901(119909)

(26)

(119904 = 1 2 3 (0 le 119909 le 2120587) for 119904 = 1 by definition1198630(119909) = 1

and 1198790= 0 0 lt 119909 lt 2120587)

At the same time both series in (26) have the well-knownrepresentations [2 542]

infin

sum

119899=1

sin 1198991199091198992119904+1

=(minus1)119904minus1

2 (2119904 + 1)(2120587)2119904+1

1198612119904+1

(119909

2120587)

((0 le 119909 le 2120587) (119904 = 1 2 3 ) 0 lt 119909 lt 2120587 for 119904 = 0)

infin

sum

119899=1

cos 1198991199091198992119904


2 (2119904)(2120587)2119904

1198612119904(119909

2120587)

(0 le 119909 le 2120587 119904 = 1 2 3 )

(27)

The previously stated procedure for obtaining (26) cannow be applied on the strength of (22)This leads to the asser-tion


infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904minus1

=(minus1)119904

21198632119904minus1

(119909)

+

119904minus1

sum

119901=1

(minus1)119901+1

1198762119904minus2119901

1198632119901minus1

(119909)

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904


21198632119904(119909)

+

119904minus1

sum

119901=0

(minus1)119901

1198762119904minus2119901

1198632119901(119909)

(28)

(119904 = 1 2 3 (minus120587 le 119909 le 120587) for 119904 = 1 by definition1198630(119909) = 1

and 1198760= 0 minus120587 lt 119909 lt 120587)


infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904+1


2 (2119904 + 1)(2120587)2119904+1

1198612119904+1

(119909 + 120587

2120587)

(minus120587 lt 119909 le 120587 119904 = 0 1 2 )

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904


2 (2119904)(2120587)2119904

1198612119904(119909 + 120587

2120587)

(minus120587 le 119909 le 120587 119904 = 1 2 3 )

(29)

By analogy with the previous the procedure for obtaining(9) (14) and (17) with the help of (8) leads us to the following


infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904minus1

=120587(minus1)

119904minus1

4[1198632119904minus2

(119909) +

119904minus1

sum

119901=1

(minus1)119901

1198702119901minus1

1198632119904minus2119901minus1

(119909)]

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904

=120587(minus1)

119904

4[1198632119904minus1

(119909) +

119904

sum

119901=1

(minus1)119901

1198702119901minus1

1198632119904minus2119901

(119909)]

(30)

119904 = 1 2 3 (0 le 119909 le 120587) for 119904 = 1 by definition 1198630(119909) =

1 (0 lt 119909 lt 120587)

At the same time both series in (30) have the well-knownformulas [2 546]

infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904+1

=(minus1)119904

1205872119904+1

4 (2119904)1198642119904(119909

120587)

(0 lt 119909 lt 120587 119904 = 0 1 2 )

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904

=(minus1)119904

1205872119904

4 (2119904 minus 1)1198642119904minus1

(119909

120587)

(0 le 119909 le 120587 119904 = 1 2 )

(31)

The same procedure applied on the base of the equality

4

120587

infin

sum

V=0

(minus1)V cos [(2V + 1) 119909]

2V + 1= 1 (minus

120587

2lt 119909 lt

120587

2) (32)

gives us the next theorem


Theorem 8 The following recursive representations holdinfin

sum

V=0

(minus1)V cos [(2V + 1) 119909](2V + 1)2119904minus1

=120587(minus1)

119904minus1

4[1198632119904minus2

(119909) +

119904minus1

sum

119901=1

(minus1)119901

11987021199011198632119904minus2119901minus2

(119909)]

infin

sum

V=0

(minus1)V sin [(2V + 1) 119909]

(2V + 1)2119904

=120587(minus1)

119904minus1

4[1198632119904minus1

(119909) +

119904minus1

sum

119901=1

(minus1)119901

11987021199011198632119904minus2119901minus1

(119909)]

(33)

(119904 = 1 2 3 (minus1205872 le 119909 le 1205872) for 119904 = 1 1198700declines

1198630(119909) = 1 (minus1205872 lt 119909 lt 1205872))

At the same time both series in (33) have the well-knownrepresentations ([2 546])infin

sum

V=0

(minus1)V cos [(2V + 1) 119909](2V + 1)2119904+1

=(minus1)119904

1205872119904+1

4 (2119904)1198642119904(119909

120587+1

2)

(minus120587

2lt 119909 lt

120587

2 119904 = 0 1 )

infin

sum

V=0

(minus1)V sin [(2V + 1) 119909]

(2V + 1)2119904=(minus1)119904minus1

1205872119904

4 (2119904 minus 1)1198642119904minus1

(119909

120587+1

2)

(minus120587

2le 119909 le

120587

2 119904 = 1 2 )

(34)

Meanwhile it is important to note that the number ofaddends in our recurrence representations (26) (28) (30)and (33) is two times less than the number of the addendsin the corresponding cited formulas from [2] So our methodappears to be more economic and effective

Moreover one can get many other representations of theconstants 119870

119903(119903 = 1 2 3 ) and numerical series 119876

2119904(119904 =

1 2 ) from Theorems 5ndash8 putting in particular 119909 = 1205872

or 119909 = 120587 For completeness we will note the main resultsFromTheorem 5 for 119909 = 1205872 and 119909 = 120587 immediately fol-

lows the following

Corollary 9 For the Favard constants119870119903 the following recur-

sive representations hold

1198702119904minus2

=4

120587(minus1)

119904

[1

21198632119904minus1

(120587

2) minus

120587

21198632119904minus2

(120587

2)]

+

119904minus1

sum

119901=1

(minus1)119901+1

1198792119904minus2119901

1198632119901minus1

(120587

2)

1198702119904minus1

=2

120587(minus1)

119904

[1

21198632119904(120587) minus

120587

21198632119904minus1

(120587)]

+

119904minus1

sum

119901=1

(minus1)119901+1

1198792119904minus2119901

1198632119901(120587)

(35)

(119904 = 1 2 for 119904 = 11198630(119909) = 1 119879

0= 0)

For 119909 = 120587 one can get (20) too by replacing previously 119904by 119904 + 1

For 119909 = 1205872 we obtain the next corollary

Corollary 10 For numbers 1198762119904 the following recursive repre-

sentations hold

1198762119904= 4119904


[1

21198632119904(120587

2) minus

120587

21198632119904minus1

(120587

2)]

+

119904minus1

sum

119901=0

(minus1)119901

1198792119904minus2119901

1198632119901(120587

2)

(36)

(119904 = 1 2 for 119904 = 11198630(119909) = 1)

Similarly from Theorem 6 for 119909 = 1205872 and 119909 = 120587 weobtain respectively the following



1198702119904minus2

=4

120587(minus1)119904minus1

21198632119904minus1

(120587

2)

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901

1198632119901minus1

(120587

2)

1198702119904minus1

=2

120587(minus1)119904minus1

21198632119904(120587)

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901

1198632119901(120587)

(37)

where 119904 = 1 2 and for 119904 = 11198630(119909) = 1 119876

0= 0


For 119909 = 1205872 from Theorem 6 (the second formula in(28)) we will have also the next analogous corollary


sentations hold

1198762119904=

4119904

4119904 minus 1(minus1)119904

21198632119904(120587

2)

+

119904minus1

sum

119901=1

(minus1)119901+1

1198762119904minus2119901

1198632119901(120587

2)

(38)

(119904 = 1 2 for 119904 = 11198630(119909) = 1 119876

0= 0)

By the same manner from Theorem 7 for 119909 = 120587 and119909 = 1205872 we obtain respectively the formulas for 119870

119903(119903 =

1 2 3 ) different from these inTheorem 1


Corollary 13 For the Favard constants 1198702119904minus3

and 1198702119904minus1

thefollowing recursive representations hold

1198702119904minus3

=(minus1)119904

1205871198632119904minus2

(120587) +

119904minus2

sum

119901=1

(minus1)119901

1198702119901minus1

1198632119904minus2119901minus1

(120587)

(39)

(119904 = 2 3 for 119904 = 2 11987011198631(120587) must be canceled) and

1198702119904minus1


1198632119904minus1

(120587

2) +

119904minus1

sum

119901=1

(minus1)119901

1198702119901minus1

1198632119904minus2119901

(120587

2)

(40)

(119904 = 1 2 3 for 119904 = 1 11987011198630(1205872)must be canceled)

The remaining cases for 119909 = 1205872 and 119909 = 120587 immediatelylead toTheorem 1 after replacing 119904 by 119904 + 1

From Theorem 8 for 119909 = 1205872 one can get respectivelyother representations for 119870

119903(119903 = 1 2 3 ) different from

these inTheorem 1

Corollary 14 For the Favard constants 119870119903 the following re-

cursive representations hold

1198702119904minus2

= (minus1)119904

1198632119904minus2

(120587

2) +

119904minus2

sum

119901=1

(minus1)119901

11987021199011198632119904minus2119901minus2

(120587

2)

1198702119904minus1


1198632119904minus1

(120587

2) +

119904minus1

sum

119901=1

(minus1)119901

11987021199011198632119904minus2119901minus1

(120587

2)

(41)

where 119904 = 1 2 3 for 119904 = 1 1198630(1205872) = minus1 and for 119904 = 2

11987021198630(1205872) must be canceled

Corollary 15 From the difference 1198792119904minus1198762119904= (1205872)119870

2119904minus1(119904 =

1 2 ) and after replacing 119904 by 119904+1 in the obtained expressionone gets the following formula

1198702119904+1

=(minus1)119904

1205871198632119904+2

(120587) +

119904

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198702119904minus2119901+1

(119904 = 0 1 )

(42)

This is somewhat better than the corresponding formula inTheorem 1 because (2119904 + 1) gt 2(2119904) for 119904 = 1 2



Here we will get down to the integral representation andrecursive formulas for the constants

119903 defined in (2) As

one can see they are closely linked with the approximationof the conjugate classes of functions obtained on the baseof the Hilbert transform [1 26] First of all let us note theirrepresentations easily obtained by means of special functions

in (3) as it is shown in [2 514] for the Catalan constant asfollows

119903=

4

120587(1 minus 2

minus(2119904+1)

) 120577 (2119904 + 1) 119903=2119904 119904 = 1 2 3

22minus4119904

120587(120577 (2119904

1

4)minus120577 (2119904

3

4))

=2

120587Γ (2119904)int

infin

0

1199052119904minus1

ch 119905119889119905 119903=2119904minus1 119904=1 2 3

(43)

In the beginning we will prove the following assertion

Theorem 16 For the constants 119903(119903 = 1 2 ) the following


2119904minus1

=2(minus1)119904

1205871198602119904minus1

(120587

2) +

119904minus1

sum

119901=1

2119904minus2119901


(2119901 minus 1)(120587

2)

2119901minus1

2119904=2(minus1)119904

1205871198602119904(120587

2) +

119904minus1

sum

119901=1

2119904minus2119901


(2119901)(120587

2)

2119901

(119904 = 1 2 3 0

def= 0)

(44)

Proof The proof of this theorem is based on induction againHowever for completeness we must give somewhat moredetailed considerations at first steps which underline furtherdiscussions

Let us start by the well-known Fourier expansions (seeeg [2 54])

infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2

= minus1

21198601(119909) (0 le 119909 le 120587)

infin

sum

119899=1

sin 1198991199091198992

= minus1198611(119909) (0 le 119909 le 2120587)

(45)

For 119909 = 1205872 the first equality gives us immediately (1205874)1= minus(12)119860

1(1205872) At the same time the second equality

in (45) leads to (1205874)1= minus1198611(1205872) Sowe obtain the integral

representation of 1in the form

1= minus

4

1205871198611(120587

2) = minus

2

1205871198601(120587

2) = 1166243616123275 sdot sdot sdot

(46)

We have another integral representation of the same con-stant

1in our paper [8]

Further by integration of both sides of the first equality in(45) we getinfin

sum

V=0

cos [(2V + 1) 119909](2V + 1)3

=1

21198602(119909) +

120587

42

(0 le 119909 le 120587)

(47)


from where for 119909 = 1205872 and 119909 = 120587 we have simultaneously0 = (12)119860

2(1205872) + (1205874)

2and minus(1205874)

2= (12)119860

2(120587) +

(1205874)2 and consequently

2= minus

2

1205871198602(120587

2) = minus

1

1205871198602(120587) (48)

Then after the integration of both sides of the secondequality in (45) we obtain

infin

sum

119899=1

cos 1198991199091198993

= 1198612(119909) + 119879

3(0 le 119909 le 2120587) (49)

from where for 119909 = 120587 we have (1205872)2= minus1198612(120587) If we cor-

respond this with (48) we find

2= minus

2

1205871198612(120587) = minus

2

1205871198602(120587

2)

= minus1

1205871198602(120587) = 1339193086109090 sdot sdot sdot

(50)

In order to obtain the integral representation of 3 we

must at first integrate the both sides of (47) as

infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)4

=1

21198603(119909) +

120587

421198631(119909) (0 le 119909 le 120587)

(51)

Next it remains to put 119909 = 1205872 as

3=2

1205871198603(120587

2) +

120587

22=2

1205871198603(120587

2) minus 119860

2(120587

2) (52)

As another integral representation of 3 we can get after

integration of both sides of (49)

infin

sum

119899=1

sin 1198991199091198994

= 1198613(119909) + 119879

31198631(119909) (0 le 119909 le 2120587) (53)

On one hand for 119909 = 120587 (53) gives

1198793= minus

1

1205871198613(120587) = 120205690 sdot sdot sdot (54)

On the other hand the same equality (53) for 119909 = 1205872

givessuminfinV=0 sin[(2V+1)1205872](2V+1)4

= 1198613(1205872)minus (12)119861

3(120587)

Then

3=4

1205871198613(120587

2) minus

2

1205871198613(120587) =

2

1205871198603(120587

2) minus 119860

2(120587

2)

= 1259163310827165 sdot sdot sdot

(55)

For the constant 4there are also different ways to receive

its integral representations One of them is based on theintegration of both sides of (53) as

minus

infin

sum

119899=1

cos 1198991199091198995

+ 1198795= 1198614(119909) + 119879

31198632(119909) (0 le 119909 le 2120587)

(56)

For 119909 = 120587 (56) gives (1205872)4= 1198614(120587)+(120587

2

2)1198793 Admit-

ting (54) we will have

4=2

1205871198614(120587) minus 119861

3(120587) = 1278999378416936 sdot sdot sdot

(57)

Another integral representation of 4can be obtained by

integration of both sides of (51) as

minus

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)5

+120587

44=1

21198604(119909) +

120587

421198632(119909)

(0 le 119909 le 120587)

(58)

For 119909 = 1205872 and 119909 = 120587 we get respectively

4=2

1205871198604(120587

2) minus

120587

41198602(120587

2) =

2

1205871198604(120587

2) +

1205872

82

=1

1205871198604(120587) +

1205872

42

(59)

Further after integration of both sides of (56) we get

minus

infin

sum

V=0

sin 1198991199091198996

+ 11987951198631(119909) = 119861

5(119909) + 119879

31198633(119909) (0 le 119909 le 2120587)

(60)

On one hand if we put here 119909 = 120587 and admit (54) we willhave

1198795=1

1205871198615(120587) +

1205872

31198793=1

1205871198615(120587) minus

120587

61198613(120587)

= 103692776 sdot sdot sdot

(61)

On the other hand the same formula (60) for 119909 = 1205872

gives us minus(1205874)5+ (1205872)119879

5= 1198615(1205872) + (120587

3

23

3)1198793 Then

admitting (54) and (61) we obtain

5= minus

120587

41198615(120587

2) +

2

1205871198615(120587) minus

120587

41198613(120587)

= 1271565517671139 sdot sdot sdot

(62)

Next in order to receive another integral representationof 5 we must integrate both sides of (58) So we will have

minus

infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)6

+120587

441198631(119909) =

1

21198602(119909) +

120587

421198633(119909)

(0 le 119909 le 120587)

(63)

Putting 119909 = 1205872 here we get

5= minus

2

1205871198605(120587

2) +

120587

24minus1205873

2332

= minus2

1205871198605(120587

2) + 119860

4(120587

2) minus

1205872

22 sdot 31198602(120587

2)

(64)

Now after this preparatory work we can go on the indi-cated procedure which leads us to the general recursive rep-resentations (43) and so complete the proof


The previously stated procedure gives us the opportunityto obtain recursive formulas for the multiple integrals119860

119903 119861119903

and 119862119903in (4) for a given 119909

First of all we will note that on the base of induction andwith the help of the first formula in (45) (47) (51) (58) and(63) we can lay down the following

Theorem 17 For themultiple singular integrals119860119903 the follow-

ing recursive representations hold

1198602119904minus1

(119909) = 2(minus1)119904

infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904

+120587

2

119904minus1

sum

119901=1

(minus1)119901

21199011198632119904minus2119901minus1

(119909)

1198602119904(119909) = minus 2(minus1)

119904

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904+1

+120587

2

119904

sum

119901=1

(minus1)119901

21199011198632119904minus2119901

(119909)

(65)

119904 = 1 2 3 (0 le 119909 le 120587) 1198630= 1 for 119904 = 0119860

0(119909) =

ln(tan(1199092)) (0 lt 119909 lt 120587) and 0= 0 where

infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904

=1

2

infin

sum

119899=1

sin 1198991199091198992119904

minus

infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905

(1198902119905

+ 1) sin119909

(1198902119905 + 1)2

minus 41198902119905cos2119909119889119905

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904+1

=1

2

infin

sum

119899=1

cos 1198991199091198992119904+1

minus

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904+1

=1

Γ (2119904+1)int

infin

0

1199052119904

119890119905

(1198902119905

minus 1) cos119909

(1198902119905+1)2

minus41198902119905cos2119909119889119905

(66)

By analogy with the previous and by means of the secondformula in (45) (49) (53) (56) and (60) one can get thefollowing

Theorem 18 For the multiple singular integrals 119861119903 the follow-


1198612119904minus1

(119909)

= (minus1)119904

infin

sum

119899=1

sin 1198991199091198992119904

+

119904minus1

sum

119901=1

(minus1)119901

1198792119904minus2119901+1

1198632119901minus1

(119909)

1198612119904minus2

(119909)

= (minus1)119904

infin

sum

119899=1

cos 1198991199091198992119904minus1

+

119904minus1

sum

119901=1

(minus1)119901

1198792119904minus2119901+1

1198632119901minus2

(119909)

(67)

119904 = 1 2 3 (0 le 119909 le 2120587) 1198630= 1 for 119904 = 1119879

1119863119896(119896 = 0 1)

declines by definition 1198610(119909) = ln(2 sin(1199092)) (0 lt 119909 lt 2120587)

where (see [2])infin

sum

119899=1

sin 1198991199091198992119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905 sin119909

1 minus 2119890119905 cos119909 + 1198902119905119889119905

(119909 = 0 119904 = 1 2 3 )

infin

sum

119899=1

cos 1198991199091198992119904minus1

=1

Γ (2119904 minus 1)int

infin

0

1199052119904minus2

(119890119905 cos119909 minus 1)

1 minus 2119890119905 cos119909 + 1198902119905119889119905

(119909 = 0 119904 = 1 2 3 )

(68)

As similar representations of 119862119903(119909) one can get on the

base of the expansion (see [2 542])infin

sum

119899=1

(minus1)119899cos 119899119909119899

= minus ln(2 cos(1199092)) (minus120587 lt 119909 lt 120587)

1198761= minus ln 2

(69)

Theorem19 For themultiple singular integrals 119862119903 the follow-


1198622119904minus1

(119909)

= (minus1)119904

infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901+1

1198632119901minus1

(119909)

1198622119904minus2

(119909)

= (minus1)119904

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904minus1

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901+1

1198632119901minus2

(119909)

(70)

119904 = 1 2 3 (minus120587 le 119909 le 120587) for 119904 = 11198620(119909) = ln (2 cos(119909

2)) (minus120587 lt 119909 lt 120587) 1198761119863119896(119896 = 0 1) declines where (see [2])

infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

=minus1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905 sin119909

1 + 2119890119905 cos119909 + 1198902119905119889119905

(119904 = 1 2 3 )

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904minus1

=minus1


infin

0

1199052119904minus2

(119890119905 cos119909 + 1)

1 + 2119890119905 cos119909 + 1198902119905119889119905

(119904 = 1 2 )

(71)

By analogy with the previous if we start from the expan-sion [2 546]infin

sum

V=0

(minus1)V sin [(2V + 1) 119909]

2V + 1=minus1

2ln(tan(120587

4minus119909

2))

(minus120587

2lt 119909 lt

120587

2)

(72)

we can obtain the next theorem


Theorem 20 The following recursive formulas hold

int

1205872minus119909

0

1198602119904minus1

(119906) 119889119906 = minus 2(minus1)119904

infin

sum

V=0

(minus1)V sin [(2V + 1) 119909](2V + 1)2119904+1

+120587

2

119904

sum

119901=1

(minus1)119901

21199011198632119904minus2119901

(120587

2minus 119909)

int

1205872minus119909

0

1198602119904minus2

(119906) 119889119906 = 2(minus1)119904

infin

sum

V=0

(minus1)V cos [(2V + 1) 119909]

(2V + 1)2119904

minus120587

2

119904minus1

sum

119901=1

(minus1)119901

21199011198632119904minus2119901minus1

(120587

2minus 119909)

(73)

119904 = 1 2 3 (minus1205872 le 119909 le 1205872) 1198630(119909) = 1 for 119904 = 1

1198600(119906) = ln tan(1199062) (minus1205872 lt 119906 lt 1205872) where

infin

sum

V=0

(minus1)V sin [(2V + 1) 119909](2V + 1)2119904+1

=1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905

(1198902119905

minus 1) sin119909

(1198902119905 + 1)2

minus 41198902119905sin2119909119889119905

(119904 = 1 2 )

infin

sum

V=0

(minus1)V cos [(2V + 1) 119909]

(2V + 1)2119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905

(1198902119905

+ 1) cos119909

(1198902119905 + 1)2

minus 41198902119905sin2119909119889119905

(119904 = 1 2 )

(74)

The proof of this theorem requires to take integrationfrom 1205872 to 119909 and then to make the substitution 119905 = 1205872 minus 119909

From Table 2 one can see that119860119903(119909) = 119861

119903(119909)minus119862

119903(119909) (0 le

119909 le 120587) 119903 = 1 2 The constants 119903satisfy the following

inequalities [1]

1 lt 1lt 3lt 5lt sdot sdot sdot lt

4

120587lt sdot sdot sdot lt

6lt 4lt 2lt120587

2

(75)

Remark 21 The equalities (49) (53) (56) and (60) outline aprocedure for summing up the numerical series 119879

2119904+1(119904 =

1 2 ) in (2) It leads us to the following


1198792119904+1

=(minus1)119904

1205871198612119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198792119904minus2119901+1

(76)

119904 = 1 2 for 119904 = 1 the term (1205872

6) 1198791declines

For comparisonwewill note also the well-known formula[2 512]

1198792119904+1

= 120577 (2119904 + 1) =1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905 minus 1119889119905

=minus1

(2119904)Ψ(2119904)

(1) (119904 = 1 2 )

(77)

The same procedure based on induction and preliminarymultiple integration of both sides of (69) leads us to the next


1198762119904+1

=(minus1)119904

1205871198622119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198762119904minus2119901+1

(78)

119904 = 1 2 for 119904 = 1 the term (1205872

6) 1198761declines

As in previous let us give the alternate formula [2 513]

1198762119904+1

= (2minus2119904

minus 1) 120577 (2119904 + 1) =minus1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905 + 1119889119905

(119904 = 1 2 )

(79)

At the end we would like to note also that one can getmany other representations (through multiple integrals) ofthe constants

119903(119903 = 1 2 ) fromTheorems 17ndash20 setting

in particular 119909 = 1205872 or 119909 = 120587 as it is made for the constants119870119903From the difference 119879

2119904+1minus 1198762119904+1

= (1205872) 2119904(119904 = 1 2

) one can get the following formula

2119904=2(minus1)119904

12058721198602119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)2119904minus2119901

(119904 = 1 2 )

(80)

which is somewhat inferior to the similar recursion formulainTheorem 16 because (2119904minus1) le 22119904minus2(2119904minus2) for 119904 = 1 2

As an exception we will give the inverse formula of (80)as

1198602119904+1

(120587) = (minus1)119904

119904

sum

119901=1

(minus1)119901+1

1205872119901

2 (2119901 minus 1)2119904minus2119901+2

(119904 = 1 2 )

(81)

If we by definition lay 0= 0 then the equality (80) is

valid for 119904 = 0 too

4 Some Notes on Numerical and ComputerImplementations of the Derived Formulas

We will consider some aspects of numerical and symboliccalculations of the Favard constants119870

119903 singular integrals (4)

and summation of series


Table 2 Calculated values of the magnitudes 119860119903(119909) 119861

119903(119909) and 119862

119903(119909) for 119909 = 1205872 120587 by using formulas (65) (67) and (70) respectively

119903 119860119903(1205872) 119861

119903(1205872) 119862

119903(1205872)

0 0346573590279972654709 0346573590279972654711 minus1831931188354438030 minus091596559417721901505 0915965594177219015062 minus2103599580529289999 minus131474973783080624966 0788849842698483749793 minus1326437390660483389 minus089924201634043412749 0427195374320049261594 minus0586164434174921923 minus041567176475313624651 0170492669421785676365 minus0200415775434410589 minus014636851588819525831 0054047259546215330636 minus0055999130148030908 minus004177075772178286763 0014228372426248040427 minus0013242890305882861 minus001003832077074076074 0003204569535142100008 minus0002716221784223860 minus000208540018388810697 0000630821600335753299 minus0000492033289602442 minus000038173093475665008 00001103023548457920610 minus0000079824623761109 minus000006247448634651347 00000173501374145951811 minus0000011727851308441 minus000000924763886476940 00000024802124436716712 minus0000001574609462951 minus000000124968150024699 000000032492796270360r 119860

119903(120587) 119861

119903(120587) 119862

119903(120587)

0 06931471805599453094171 0 0 02 minus42071991610585799989 minus210359958052928999945 2103599580529289999453 minus66086529882853881226 minus377637313616307892720 2832279852122309195404 minus63627527878802461560 minus392286552530160912232 2439887262578637033685 minus45591894893017946564 minus295428020294335768590 1604909286358436970496 minus26255391709472096141 minus176271265891422894113 0862826512032980672957 minus12684915655900274347 minus087472015815662857450 0393771407433398860168 minus05287656648068309189 minus037237211738940254175 0156393547417428377149 minus01940031559506448617 minus013897143291435930434 00550317230362855573910 minus00635985732918145806 minus004620733330628432785 00173912399855302527711 minus00188489390019113804 minus001385968931040814707 00049892496915032333612 minus00050985327885974909 minus000378779048977775268 000131074229881973820

Let us take only the first119898minus1119898 le 119904+1 terms in the finalsums in the right-hand side of the formulas (7a) and (7b) anddenote the remaining truncation sums by

1198781119898

=

119904

sum

119895=119898


120587119895119901

(2119895)1198702119904minus2119895+1

1198782119898

=

119904

sum

119895=119898


(2119895 minus 1)(120587

2)

2119895minus1

1198702119904minus2119895+1

(82)

respectively

Theorem 24 For 119898 ge 2 and any 119904 gt 119898 the following esti-mates for the truncation errors (82) hold

100381610038161003816100381611987811198981003816100381610038161003816 = 119874(

1205872119898

(2119898))

100381610038161003816100381611987821198981003816100381610038161003816 = 119874((

120587

2)

2119898minus11

(2119898 minus 1))

(83)

where Landau big119874 notation is used For the Landau notationsee [27]

Proof By means of the inequalities (25) it is easy to obtain

100381610038161003816100381611987811198981003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119904

sum

119895=119898


120587119895119901

(2119895)1198702119904minus2119895+1

10038161003816100381610038161003816100381610038161003816100381610038161003816

le120587

2

119904

sum

119895=119898

1205872119895

(2119895)=120587

2

1205872119898

(2119898)

times (1 +1205872

(2119898 + 1) (2119898 + 2)+ sdot sdot sdot +

1205872119904minus2119898

(2119898 + 1) sdot sdot sdot (2119904))

le1205872119898+1

2 (2119898)(1 +

10

56+102

5262+ sdot sdot sdot )

le1205872119898+1

2 (2119898)(1 +

1

3+1


=1205872119898+1

2 (2119898)

3

2le31205872119898

(2119898)

(84)


m = 12 Array [K m] K1=120587

2 d [n

minus xminus] =

xn

n

For [s = 1 s le m2 s + +

K2s+1 =

1

2((minus1)

sd [2s + 1 120587] +s

sum

p=1

K2s+1minus2p

(minus1)pminus11205872p

(2p))

K2s = (minus1)

sd [2s120587

2] +

s

sum

p=1

K2s+1minus2p

(minus1)pminus1(120587

2)

2pminus1

(2p minus 1)

]

For [s = 1 s le m s ++ Print [Ks N[Ks 30]]]

Algorithm 1Mathematica code for exact symbolic and approximate computation with optional 30 digits accuracy of the Favard constantsby recursive representations (7a) and (7b)

Consequently |1198781119898| = 119874(120587

2119898

(2119898)) In the sameway forthe second truncation sum we have

100381610038161003816100381611987821198981003816100381610038161003816 le 2(

120587

2)

2119898minus11

(2119898 minus 1)

so 100381610038161003816100381611987821198981003816100381610038161003816 = 119874((

120587

2)

2119898minus11

(2119898 minus 1))

(85)

By the details of the proof it follows that the obtainedrepresentations did not depend on 119904 for any 119898 lt 119904 The apriori estimates (83) can be used for calculation of 119870

119903with a

given numerical precision 120576 in order to decrease the numberof addends in the sums for larger 119904 at the condition119898 lt 119904

Remark 25 The error estimates similar to (83) can be estab-lished for other recurrence representations in this paper too

In Algorithm 1 we provide the simplest Mathematicacode for exact symbolic and numerical calculation withoptional 30 digits accuracy of the Favard constants 119870

119903for

119903 = 1 2 3 119898 based on recursive representations (7a) and(7b) for a given arbitrary integer119898 gt 0 The obtained resultsare given in Table 1 We have to note that this code is not themost economic It can be seen that the thrifty code will takeabout 21198982 + 8119898 or 119874(1198982) arithmetic operations in (7a) and(7b)

The basic advantage of using formulas (7a) and (7b) is thatthey contain only finite number of terms (ie finite number ofarithmetic operations) in comparisonwith the initial formula119870119903= (4120587)sum

infin

]=0((minus1)](119903+1)

(2] + 1)119903+1

) (119903 = 0 1 2 )which needs the calculation of the slowly convergent infinitesum It must be also mentioned that inMathematicaMapleand other powerful mathematical software packages theFavard constants are represented by sums of Zeta and relatedfunctions which are calculated by the use of Euler-Maclaurinsummation and functional equations Near the critical stripthey also use the Riemann-Siegel formula (see eg [28A94])

Finally we will note that the direct numerical integrationin (4) is very difficult This way a big opportunity is given

by Theorems 17 18 and 19 (formulas (65) (67) and (70))for effective numerical calculation of the classes of multiplesingular integrals 119860

119903(119909) 119861

119903(119909) and 119862

119903(119909) for any 119909 Their

computation is reduced to find the finite number of theFavard constants V for all V le 119903 and the calculation of oneadditional numerical sum Numerical values for some of thesingular integrals are presented in Table 2

5 Concluding Remarks

As it became clear there are many different ways and well-developed computer programs at present for calculation ofthe significant constants inmathematics (in particular Favardconstants) and for summation of important numerical seriesMost of them are based on using of generalized functions asone can see in [2ndash5]Nonetheless we hope that our previouslystated approach through integration of Fourier series appearsto be more convenient and has its theoretical and practicalmeanings in the scope of applications in particular forcomputing of the pointed special types of multiple singularintegrals The basic result with respect to multiple integralsreduces their calculation to finite number of numerical sums

Acknowledgments

This paper is supported by the Bulgarian Ministry of Edu-cation Youth and Science Grant BG051PO00133-05-0001ldquoScience and businessrdquo financed under operational programldquoHuman Resources Developmentrdquo by the European SocialFund

References

[1] N Korneıchuk Exact Constants in Approximation Theory vol38 chapter 3 4 Cambridge University Press New York NYUSA 1991

[2] A P Prudnikov A Yu Brychkov and O I Marichev Integralsand Series Elementary Functions chapter 5 CRC Press BocaRaton Fla USA 1998

[3] J Favard ldquoSur les meilleurs precedes drsquoapproximation de cer-taines classes de fonctions par des polynomes trigonometri-quesrdquo Bulletin des Sciences Mathematiques vol 61 pp 209ndash2241937


[4] S R Finch Mathematical Constants Cambridge UniversityPress New York NY USA 2003

[5] J Bustamante Algebraic Approximation A Guide to Past andCurrent Solutions Springer Basel AG Basel Switzerland 2012

[6] S Foucart Y Kryakin and A Shadrin ldquoOn the exact constantin the Jackson-Stechkin inequality for the uniform metricrdquoConstructive Approximation vol 29 no 2 pp 157ndash179 2009

[7] Yu N Subbotin and S A Telyakovskiı ldquoOn the equality ofKolmogorov and relative widths of classes of differentiablefunctionsrdquoMatematicheskie Zametki vol 86 no 3 pp 432ndash4392009

[8] I H Feschiev and S G Gocheva-Ilieva ldquoOn the extension ofa theorem of Stein and Weiss and its applicationrdquo ComplexVariables vol 49 no 10 pp 711ndash730 2004

[9] R A DeVore and G G Lorentz Constructive Approximationvol 303 Springer Berlin Germany 1993

[10] V P Motornyı ldquoOn sharp estimates for the pointwise approx-imation of the classes 119882

119903

119867120596 by algebraic polynomialsrdquo

Ukrainian Mathematical Journal vol 53 no 6 pp 916ndash9372001 Translation fromUkrainsrsquokyiMatematychnyi Zhurnal vol53 no 6 pp 783ndash799 2001

[11] W Xiao ldquoRelative infinite-dimensional width of Sobolev classes119882119903

119901(119877)rdquo Journal of Mathematical Analysis and Applications vol

369 no 2 pp 575ndash582 2010[12] P C Nitiema ldquoOn the best one-sided approximation of func-

tions in themeanrdquo Far East Journal of AppliedMathematics vol49 no 2 pp 139ndash150 2010

[13] O L Vinogradov ldquoSharp inequalities for approximations ofclasses of periodic convolutions by subspaces of shifts of odddimensionrdquo Mathematical Notes vol 85 no 4 pp 544ndash5572009 Translation fromMatematicheskie Zametki vol 85 no 4pp 569ndash584 2009

[14] A V Mironenko ldquoOn the Jackson-Stechkin inequality foralgebraic polynomialsrdquo Proceedings of Institute of Mathematicsand Mechanics vol 273 supplement 1 pp S116ndashS123 2011

[15] V F Babenko and V A Zontov ldquoBernstein-type inequalitiesfor splines defined on the real axisrdquo Ukrainian MathematicalJournal vol 63 pp 699ndash708 2011

[16] G Vainikko ldquoError estimates for the cardinal spline interpola-tionrdquo Journal of Analysis and its Applications vol 28 no 2 pp205ndash222 2009

[17] O L Vinogradov ldquoAnalog of the Akhiezer-Krein-Favard sumsfor periodic splines of minimal defectrdquo Journal of MathematicalSciences vol 114 no 5 pp 1608ndash1627 2003

[18] L A Apaıcheva ldquoOptimal quadrature and cubature formulasfor singular integrals with Hilbert kernelsrdquo Izvestiya VysshikhUchebnykh Zavedeniı no 4 pp 14ndash25 2004

[19] F D Gakhov and I Kh Feschiev ldquoApproximate calculationof singular integralsrdquo Izvestiya Akademii Nauk BSSR SeriyaFiziko-Matematicheskikh Nauk vol 4 pp 5ndash12 1977

[20] F D Gakhov and I Kh Feschiev ldquoInterpolation of singularintegrals and approximate solution of the Riemann boundaryvalue problemrdquo Vestsı Akademıı Navuk BSSR Seryya Fızıka-Matematychnykh Navuk no 5 pp 3ndash13 1982

[21] B G Gabdulkhaev ldquoFinite-dimensional approximations of sin-gular integrals and direct methods of solution of singular inte-gral and integro-differential equationsrdquo Journal of Soviet Math-ematics vol 18 pp 593ndash627 1982

[22] V P Motornyı ldquoApproximation of some classes of singularintegrals by algebraic polynomialsrdquo Ukrainian MathematicalJournal vol 53 no 3 pp 377ndash394 2001

[23] E Vainikko and G Vainikko ldquoProduct quasi-interpolationin logarithmically singular integral equationsrdquo MathematicalModelling and Analysis vol 17 no 5 pp 696ndash714 2012

[24] H Brass and K Petras Quadrature Theory The Theory ofNumerical Integration on a Compact Interval vol 178 AmericanMathematical Society Providence RI USA 2011

[25] E W Weisstein ldquoFavard constantsrdquo httpmathworldwolframcomFavardConstantshtml

[26] A G ZygmundTrigonometric Series vol 1 Cambridge Univer-sity Press 2nd edition 1959

[27] D E Knuth The Art of Computer Programming Vol 1 Funda-mental Algorithms Section 1211 Asymptotic RepresentationsAddison-Wesley 3rd edition 1997

[28] S Wolfram The Mathematica Book Wolfram Media Cham-paign Ill USA 5th edition 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

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Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


120577 (119911 119886) =

infin

sum

119896=0

1

(119896 + 119886)119911=

1

Γ (119911)int

infin

0

119905119911minus1

119890minus119886119905

1 minus 119890minus119905119889119905 (Re 119911 gt 1)

119861119899(119909)

119905119890119909119905

119890119905 minus 1=

infin

sum

119896=0

119861119896(119909)

119905119896

119896(|119905| lt 2120587)

119861119899= 119861119899(0) 119861

119899(119909) =

119899

sum

119896=0

(119899

119896)119861119896119909119899minus119896

119864119899(119909)

2119890119909119905

119890119905 + 1=

infin

sum

119896=0

119864119896(119909)

119905119896

119896

(1

cosh (119905)=

infin

sum

119896=0

119864119896

119905119896

119896) (|119905| lt 120587)

(3)

From now on we will use the following notations

119860119899(119909) = int

119909

0


119909

0

ln tan 1205792119889120579 (0 le 119909 le 120587)

119861119899(119909) = int

119909

0


119909

0

ln(2 sin 1205792) 119889120579 (0 le 119909 le 2120587)

119862119899(119909) = int

119909

0


119909

0

ln(2 cos 1205792) 119889120579

(minus120587 le 119909 le 120587) (119899 = 1 2 3 )

(4)

119863119899(119909) =

119909119899

119899 (5)

119879119903=

infin

sum

119899=1

1

119899119903(119903 = 2 3 )

119876119903=

infin

sum

119899=1

(minus1)119899

119899119903(119903 = 1 2 3 )

(6)

where the threemultiple singular integrals (4) contain exactly119899 integral operations

The main purpose of this paper is to establish newrecursive formulas for the Favard constants (1) including onlyfinite number of terms and recursive formulas for (2) Theywill be further used to obtain new integral representations forthe previously stated objects in particular for calculation ofthe multiple singular integrals (4) and summation of series



We will prove the following

Theorem 1 For the constants119870119903(119903 = 1 2 3 ) the following


1198702119904+1

=1

2[(minus1)

119904

1198632119904+1

(120587) +

119904

sum

119901=1

1198702119904+1minus2119901


1205872119901

(2119901)]

(1198700= 1 119870

1=120587

2)

(7a)

1198702119904= (minus1)

119904

1198632119904(120587

2) +

119904

sum

119901=1

1198702119904+1minus2119901


(2119901 minus 1)(120587

2)

2119901minus1

(119904 = 1 2 3 )

(7b)

Proof We will use the method of induction and preliminaryintegration of appropriate Fourier series Let us start by thewell-known expansion [2]

4

120587

infin

sum

]=0

sin [(2] + 1) 119909]2] + 1

= 1 (0 lt 119909 lt 120587) (8)

where for 119909 = 1205872we will have1198700= 1 By integration of both

sides of (8) in [0 119909] we get

minus4

120587

infin

sum

]=0

cos [(2] + 1) 119909](2] + 1)2

+4

120587

infin

sum

]=0

1

(2] + 1)2= 119909 = 119863

1(119909)

(0 le 119909 le 120587)

(9)

For 119909 = 120587 (9) gives us 21198701= 120587 = 119863

1(120587) and conse-

quently1198701= 1198631(120587)2 = 1205872 The same results for119870

0and119870

1

can be achieved starting from the equality [2]

infin

sum

119899=1

sin 119899119909119899

=120587

2minus119909

2=120587

2minus1

21198631(119909) (0 lt 119909 lt 2120587)

(10)

For 119909 = 1205872 we find again that 1198700= 1 After integration

of the both sides of (10) we have

minus

infin

sum

119899=1

cos 1198991199091198992

+ 1198792= minus

1199092

4+120587119909

2= minus

1

21198632(119909) +

120587

21198631(119909)

(0 le 119909 le 2120587)

(11)

For 119909 = 120587 we find that 1198701= 1205872 Next after integration

of the both sides of (11) we obtain

minus

infin

sum

119899=1

sin 1198991199091198993

+ 1198792119909 = minus

1

21198633(119909) +

120587

21198632(119909) (0 le 119909 le 2120587)

(12)

which for 119909 = 120587 gives us

1198792= minus

1

21205871198633(120587) +

1

21198632(120587) =

1205872

6 (13)



2= 1205872


we obtain

minus4

120587

infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)3

+ 1198701119909 = 119863

2(119909) (0 le 119909 le 120587)

(14)


1= 1198632(1205872) and

consequently

1198702=120587

21198701minus 1198632(120587

2) =

1205872

8 (15)


(12) asinfin

sum

119899=1

cos 1198991199091198994

minus 1198794+ 11987921198632(119909) = minus

1

21198634(119909) +

120587

21198633(119909)

(0 le 119909 le 2120587)

(16)


4

120587

infin

sum

V=1

cos [(2V + 1) 119909](2V + 1)4

minus4

120587

infin

sum

V=1

1

(2V + 1)4+ 11987011198632(119909) = 119863

3(119909)

(0 le 119909 le 120587)

(17)


1198703=1

1205871198634(120587) minus

3

21198633(120587) +

120587

21198632(120587)

=1205872

41198701minus1

21198633(120587) =

1205873

24

(18)






1198702119904minus1

=2

120587 (2119904)(22119904

minus 1) 1205872119904 10038161003816100381610038161198612119904

1003816100381610038161003816

1198702119904=

2

120587 (2119904)(120587

2)

2119904+1

100381610038161003816100381611986421199041003816100381610038161003816 (119904 = 1 2 3 )

(19)





the assertion



119903 Exact values of119870119903


1 1205872 157079632679489661923132169162 120587

2

8 123370055013616982735431137493 120587

3

24 129192819501249250731151312774 5120587

4

384 126834753950524006818281683185 120587

5

240 127508201993867272192808879186 61120587

6

46080 127267232656453061325614987117 17120587

7

40320 127343712480668316338644619008 277120587

8

2064384 127317548065260581363477696719 31120587

9

725760 1273261242472487546381436665610 50521120587

10

3715891200 1273232382729394849508279710811 691120587

11

159667200 1273241945872154096771507790112 540553120587

12

392398110720 12732387471572495304117396905


1198792119904= (minus1)

119904

[1

21205871198632119904+1

(120587) minus1

21198632119904(120587)]

+

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198792119904minus2119901

(119904 = 1 2 3 )

(20)




1198792119904=22119904minus1

1205872119904

(2119904)

100381610038161003816100381611986121199041003816100381610038161003816 (119904 = 1 2 3 ) (21)


sum

119899=1

(minus1)119899sin 119899119909119899

= minus1

21198631(119909) (minus120587 lt 119909 lt 120587) (22)



1198762119904=(minus1)119904

21205871198632119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198762119904minus2119901

(119904 = 1 2 3 )

(23)



1198762119904=


) 1205872119904

(2119904)

100381610038161003816100381611986121199041003816100381610038161003816 (119904 = 1 2 3 ) (24)




4


5lt 1198703lt 1198701=120587

2

(25)


119870119903= 4120587




infin

sum

119899=1

sin 1198991199091198992119904minus1

= (minus1)119904

[1

21198632119904minus1

(119909) minus120587

21198632119904minus2

(119909)]

+

119904minus1

sum

119901=1

(minus1)119901+1

1198792119904minus2119901

1198632119901minus1

(119909)

infin

sum

119899=1

cos 1198991199091198992119904


[1

21198632119904(119909) minus

120587

21198632119904minus1

(119909)]

+

119904minus1

sum

119901=0

(minus1)119901

1198792119904minus2119901

1198632119901(119909)

(26)


and 1198790= 0 0 lt 119909 lt 2120587)


infin

sum

119899=1

sin 1198991199091198992119904+1


2 (2119904 + 1)(2120587)2119904+1

1198612119904+1

(119909

2120587)

((0 le 119909 le 2120587) (119904 = 1 2 3 ) 0 lt 119909 lt 2120587 for 119904 = 0)

infin

sum

119899=1

cos 1198991199091198992119904


2 (2119904)(2120587)2119904

1198612119904(119909

2120587)

(0 le 119909 le 2120587 119904 = 1 2 3 )

(27)



infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904minus1

=(minus1)119904

21198632119904minus1

(119909)

+

119904minus1

sum

119901=1

(minus1)119901+1

1198762119904minus2119901

1198632119901minus1

(119909)

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904


21198632119904(119909)

+

119904minus1

sum

119901=0

(minus1)119901

1198762119904minus2119901

1198632119901(119909)

(28)


and 1198760= 0 minus120587 lt 119909 lt 120587)


infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904+1


2 (2119904 + 1)(2120587)2119904+1

1198612119904+1

(119909 + 120587

2120587)

(minus120587 lt 119909 le 120587 119904 = 0 1 2 )

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904


2 (2119904)(2120587)2119904

1198612119904(119909 + 120587

2120587)

(minus120587 le 119909 le 120587 119904 = 1 2 3 )

(29)



infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904minus1

=120587(minus1)

119904minus1

4[1198632119904minus2

(119909) +

119904minus1

sum

119901=1

(minus1)119901

1198702119901minus1

1198632119904minus2119901minus1

(119909)]

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904

=120587(minus1)

119904

4[1198632119904minus1

(119909) +

119904

sum

119901=1

(minus1)119901

1198702119901minus1

1198632119904minus2119901

(119909)]

(30)


1 (0 lt 119909 lt 120587)


infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904+1

=(minus1)119904

1205872119904+1

4 (2119904)1198642119904(119909

120587)

(0 lt 119909 lt 120587 119904 = 0 1 2 )

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904

=(minus1)119904

1205872119904

4 (2119904 minus 1)1198642119904minus1

(119909

120587)

(0 le 119909 le 120587 119904 = 1 2 )

(31)


4

120587

infin

sum

V=0

(minus1)V cos [(2V + 1) 119909]

2V + 1= 1 (minus

120587

2lt 119909 lt

120587

2) (32)




sum

V=0


=120587(minus1)

119904minus1

4[1198632119904minus2

(119909) +

119904minus1

sum

119901=1

(minus1)119901

11987021199011198632119904minus2119901minus2

(119909)]

infin

sum

V=0

(minus1)V sin [(2V + 1) 119909]

(2V + 1)2119904

=120587(minus1)

119904minus1

4[1198632119904minus1

(119909) +

119904minus1

sum

119901=1

(minus1)119901

11987021199011198632119904minus2119901minus1

(119909)]

(33)


1198630(119909) = 1 (minus1205872 lt 119909 lt 1205872))


sum

V=0

(minus1)V cos [(2V + 1) 119909](2V + 1)2119904+1

=(minus1)119904

1205872119904+1

4 (2119904)1198642119904(119909

120587+1

2)

(minus120587

2lt 119909 lt

120587

2 119904 = 0 1 )

infin

sum

V=0

(minus1)V sin [(2V + 1) 119909]

(2V + 1)2119904=(minus1)119904minus1

1205872119904

4 (2119904 minus 1)1198642119904minus1

(119909

120587+1

2)

(minus120587

2le 119909 le

120587

2 119904 = 1 2 )

(34)




2119904(119904 =



lows the following



1198702119904minus2

=4

120587(minus1)

119904

[1

21198632119904minus1

(120587

2) minus

120587

21198632119904minus2

(120587

2)]

+

119904minus1

sum

119901=1

(minus1)119901+1

1198792119904minus2119901

1198632119901minus1

(120587

2)

1198702119904minus1

=2

120587(minus1)

119904

[1

21198632119904(120587) minus

120587

21198632119904minus1

(120587)]

+

119904minus1

sum

119901=1

(minus1)119901+1

1198792119904minus2119901

1198632119901(120587)

(35)

(119904 = 1 2 for 119904 = 11198630(119909) = 1 119879

0= 0)




sentations hold

1198762119904= 4119904


[1

21198632119904(120587

2) minus

120587

21198632119904minus1

(120587

2)]

+

119904minus1

sum

119901=0

(minus1)119901

1198792119904minus2119901

1198632119901(120587

2)

(36)

(119904 = 1 2 for 119904 = 11198630(119909) = 1)




1198702119904minus2

=4

120587(minus1)119904minus1

21198632119904minus1

(120587

2)

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901

1198632119901minus1

(120587

2)

1198702119904minus1

=2

120587(minus1)119904minus1

21198632119904(120587)

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901

1198632119901(120587)

(37)

where 119904 = 1 2 and for 119904 = 11198630(119909) = 1 119876

0= 0




sentations hold

1198762119904=

4119904

4119904 minus 1(minus1)119904

21198632119904(120587

2)

+

119904minus1

sum

119901=1

(minus1)119901+1

1198762119904minus2119901

1198632119901(120587

2)

(38)

(119904 = 1 2 for 119904 = 11198630(119909) = 1 119876

0= 0)


119903(119903 =




and 1198702119904minus1


1198702119904minus3

=(minus1)119904

1205871198632119904minus2

(120587) +

119904minus2

sum

119901=1

(minus1)119901

1198702119901minus1

1198632119904minus2119901minus1

(120587)

(39)


1198702119904minus1


1198632119904minus1

(120587

2) +

119904minus1

sum

119901=1

(minus1)119901

1198702119901minus1

1198632119904minus2119901

(120587

2)

(40)




119903(119903 = 1 2 3 ) different from

these inTheorem 1



1198702119904minus2

= (minus1)119904

1198632119904minus2

(120587

2) +

119904minus2

sum

119901=1

(minus1)119901

11987021199011198632119904minus2119901minus2

(120587

2)

1198702119904minus1


1198632119904minus1

(120587

2) +

119904minus1

sum

119901=1

(minus1)119901

11987021199011198632119904minus2119901minus1

(120587

2)

(41)


11987021198630(1205872) must be canceled


2119904minus1(119904 =


1198702119904+1

=(minus1)119904

1205871198632119904+2

(120587) +

119904

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198702119904minus2119901+1

(119904 = 0 1 )

(42)








119903=

4

120587(1 minus 2

minus(2119904+1)

) 120577 (2119904 + 1) 119903=2119904 119904 = 1 2 3

22minus4119904

120587(120577 (2119904

1

4)minus120577 (2119904

3

4))

=2

120587Γ (2119904)int

infin

0

1199052119904minus1

ch 119905119889119905 119903=2119904minus1 119904=1 2 3

(43)




2119904minus1

=2(minus1)119904

1205871198602119904minus1

(120587

2) +

119904minus1

sum

119901=1

2119904minus2119901


(2119901 minus 1)(120587

2)

2119901minus1

2119904=2(minus1)119904

1205871198602119904(120587

2) +

119904minus1

sum

119901=1

2119904minus2119901


(2119901)(120587

2)

2119901

(119904 = 1 2 3 0

def= 0)

(44)



infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2

= minus1

21198601(119909) (0 le 119909 le 120587)

infin

sum

119899=1

sin 1198991199091198992

= minus1198611(119909) (0 le 119909 le 2120587)

(45)





1= minus

4

1205871198611(120587

2) = minus

2

1205871198601(120587

2) = 1166243616123275 sdot sdot sdot

(46)


1in our paper [8]


sum

V=0

cos [(2V + 1) 119909](2V + 1)3

=1

21198602(119909) +

120587

42

(0 le 119909 le 120587)

(47)



2(1205872) + (1205874)

2and minus(1205874)

2= (12)119860

2(120587) +


2= minus

2

1205871198602(120587

2) = minus

1

1205871198602(120587) (48)


infin

sum

119899=1

cos 1198991199091198993

= 1198612(119909) + 119879

3(0 le 119909 le 2120587) (49)



2= minus

2

1205871198612(120587) = minus

2

1205871198602(120587

2)

= minus1

1205871198602(120587) = 1339193086109090 sdot sdot sdot

(50)



infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)4

=1

21198603(119909) +

120587

421198631(119909) (0 le 119909 le 120587)

(51)


3=2

1205871198603(120587

2) +

120587

22=2

1205871198603(120587

2) minus 119860

2(120587

2) (52)



infin

sum

119899=1

sin 1198991199091198994

= 1198613(119909) + 119879

31198631(119909) (0 le 119909 le 2120587) (53)


1198793= minus

1

1205871198613(120587) = 120205690 sdot sdot sdot (54)



= 1198613(1205872)minus (12)119861

3(120587)

Then

3=4

1205871198613(120587

2) minus

2

1205871198613(120587) =

2

1205871198603(120587

2) minus 119860

2(120587

2)

= 1259163310827165 sdot sdot sdot

(55)



minus

infin

sum

119899=1

cos 1198991199091198995

+ 1198795= 1198614(119909) + 119879

31198632(119909) (0 le 119909 le 2120587)

(56)

For 119909 = 120587 (56) gives (1205872)4= 1198614(120587)+(120587

2

2)1198793 Admit-


4=2

1205871198614(120587) minus 119861

3(120587) = 1278999378416936 sdot sdot sdot

(57)



minus

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)5

+120587

44=1

21198604(119909) +

120587

421198632(119909)

(0 le 119909 le 120587)

(58)


4=2

1205871198604(120587

2) minus

120587

41198602(120587

2) =

2

1205871198604(120587

2) +

1205872

82

=1

1205871198604(120587) +

1205872

42

(59)


minus

infin

sum

V=0

sin 1198991199091198996

+ 11987951198631(119909) = 119861

5(119909) + 119879

31198633(119909) (0 le 119909 le 2120587)

(60)


1198795=1

1205871198615(120587) +

1205872

31198793=1

1205871198615(120587) minus

120587

61198613(120587)


(61)


gives us minus(1205874)5+ (1205872)119879

5= 1198615(1205872) + (120587

3

23

3)1198793 Then


5= minus

120587

41198615(120587

2) +

2

1205871198615(120587) minus

120587

41198613(120587)

= 1271565517671139 sdot sdot sdot

(62)


minus

infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)6

+120587

441198631(119909) =

1

21198602(119909) +

120587

421198633(119909)

(0 le 119909 le 120587)

(63)


5= minus

2

1205871198605(120587

2) +

120587

24minus1205873

2332

= minus2

1205871198605(120587

2) + 119860

4(120587

2) minus

1205872

22 sdot 31198602(120587

2)

(64)




119903 119861119903

and 119862119903in (4) for a given 119909




1198602119904minus1

(119909) = 2(minus1)119904

infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904

+120587

2

119904minus1

sum

119901=1

(minus1)119901

21199011198632119904minus2119901minus1

(119909)

1198602119904(119909) = minus 2(minus1)

119904

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904+1

+120587

2

119904

sum

119901=1

(minus1)119901

21199011198632119904minus2119901

(119909)

(65)

119904 = 1 2 3 (0 le 119909 le 120587) 1198630= 1 for 119904 = 0119860

0(119909) =


infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904

=1

2

infin

sum

119899=1

sin 1198991199091198992119904

minus

infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905

(1198902119905

+ 1) sin119909

(1198902119905 + 1)2

minus 41198902119905cos2119909119889119905

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904+1

=1

2

infin

sum

119899=1

cos 1198991199091198992119904+1

minus

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904+1

=1

Γ (2119904+1)int

infin

0

1199052119904

119890119905

(1198902119905

minus 1) cos119909

(1198902119905+1)2

minus41198902119905cos2119909119889119905

(66)




1198612119904minus1

(119909)

= (minus1)119904

infin

sum

119899=1

sin 1198991199091198992119904

+

119904minus1

sum

119901=1

(minus1)119901

1198792119904minus2119901+1

1198632119901minus1

(119909)

1198612119904minus2

(119909)

= (minus1)119904

infin

sum

119899=1

cos 1198991199091198992119904minus1

+

119904minus1

sum

119901=1

(minus1)119901

1198792119904minus2119901+1

1198632119901minus2

(119909)

(67)

119904 = 1 2 3 (0 le 119909 le 2120587) 1198630= 1 for 119904 = 1119879

1119863119896(119896 = 0 1)



sum

119899=1

sin 1198991199091198992119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905 sin119909

1 minus 2119890119905 cos119909 + 1198902119905119889119905

(119909 = 0 119904 = 1 2 3 )

infin

sum

119899=1

cos 1198991199091198992119904minus1

=1


infin

0

1199052119904minus2

(119890119905 cos119909 minus 1)

1 minus 2119890119905 cos119909 + 1198902119905119889119905

(119909 = 0 119904 = 1 2 3 )

(68)



sum

119899=1

(minus1)119899cos 119899119909119899


1198761= minus ln 2

(69)



1198622119904minus1

(119909)

= (minus1)119904

infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901+1

1198632119901minus1

(119909)

1198622119904minus2

(119909)

= (minus1)119904

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904minus1

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901+1

1198632119901minus2

(119909)

(70)



infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

=minus1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905 sin119909

1 + 2119890119905 cos119909 + 1198902119905119889119905

(119904 = 1 2 3 )

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904minus1

=minus1


infin

0

1199052119904minus2

(119890119905 cos119909 + 1)

1 + 2119890119905 cos119909 + 1198902119905119889119905

(119904 = 1 2 )

(71)


sum

V=0

(minus1)V sin [(2V + 1) 119909]

2V + 1=minus1

2ln(tan(120587

4minus119909

2))

(minus120587

2lt 119909 lt

120587

2)

(72)




int

1205872minus119909

0

1198602119904minus1

(119906) 119889119906 = minus 2(minus1)119904

infin

sum

V=0

(minus1)V sin [(2V + 1) 119909](2V + 1)2119904+1

+120587

2

119904

sum

119901=1

(minus1)119901

21199011198632119904minus2119901

(120587

2minus 119909)

int

1205872minus119909

0

1198602119904minus2

(119906) 119889119906 = 2(minus1)119904

infin

sum

V=0

(minus1)V cos [(2V + 1) 119909]

(2V + 1)2119904

minus120587

2

119904minus1

sum

119901=1

(minus1)119901

21199011198632119904minus2119901minus1

(120587

2minus 119909)

(73)

119904 = 1 2 3 (minus1205872 le 119909 le 1205872) 1198630(119909) = 1 for 119904 = 1


infin

sum

V=0

(minus1)V sin [(2V + 1) 119909](2V + 1)2119904+1

=1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905

(1198902119905

minus 1) sin119909

(1198902119905 + 1)2

minus 41198902119905sin2119909119889119905

(119904 = 1 2 )

infin

sum

V=0

(minus1)V cos [(2V + 1) 119909]

(2V + 1)2119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905

(1198902119905

+ 1) cos119909

(1198902119905 + 1)2

minus 41198902119905sin2119909119889119905

(119904 = 1 2 )

(74)



119903(119909)minus119862

119903(119909) (0 le


inequalities [1]


4


6lt 4lt 2lt120587

2

(75)


2119904+1(119904 =



1198792119904+1

=(minus1)119904

1205871198612119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198792119904minus2119901+1

(76)

119904 = 1 2 for 119904 = 1 the term (1205872

6) 1198791declines


1198792119904+1

= 120577 (2119904 + 1) =1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905 minus 1119889119905

=minus1

(2119904)Ψ(2119904)

(1) (119904 = 1 2 )

(77)



1198762119904+1

=(minus1)119904

1205871198622119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198762119904minus2119901+1

(78)

119904 = 1 2 for 119904 = 1 the term (1205872

6) 1198761declines


1198762119904+1

= (2minus2119904

minus 1) 120577 (2119904 + 1) =minus1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905 + 1119889119905

(119904 = 1 2 )

(79)




2119904+1minus 1198762119904+1

= (1205872) 2119904(119904 = 1 2


2119904=2(minus1)119904

12058721198602119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)2119904minus2119901

(119904 = 1 2 )

(80)



1198602119904+1

(120587) = (minus1)119904

119904

sum

119901=1

(minus1)119901+1

1205872119901

2 (2119901 minus 1)2119904minus2119901+2

(119904 = 1 2 )

(81)









119903(119909) and 119862


119903 119860119903(1205872) 119861

119903(1205872) 119862

119903(1205872)


119903(120587) 119861

119903(120587) 119862

119903(120587)



1198781119898

=

119904

sum

119895=119898


120587119895119901

(2119895)1198702119904minus2119895+1

1198782119898

=

119904

sum

119895=119898


(2119895 minus 1)(120587

2)

2119895minus1

1198702119904minus2119895+1

(82)

respectively


100381610038161003816100381611987811198981003816100381610038161003816 = 119874(

1205872119898

(2119898))

100381610038161003816100381611987821198981003816100381610038161003816 = 119874((

120587

2)

2119898minus11

(2119898 minus 1))

(83)



100381610038161003816100381611987811198981003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119904

sum

119895=119898


120587119895119901

(2119895)1198702119904minus2119895+1

10038161003816100381610038161003816100381610038161003816100381610038161003816

le120587

2

119904

sum

119895=119898

1205872119895

(2119895)=120587

2

1205872119898

(2119898)

times (1 +1205872

(2119898 + 1) (2119898 + 2)+ sdot sdot sdot +

1205872119904minus2119898

(2119898 + 1) sdot sdot sdot (2119904))

le1205872119898+1

2 (2119898)(1 +

10

56+102


le1205872119898+1

2 (2119898)(1 +

1

3+1


=1205872119898+1

2 (2119898)

3

2le31205872119898

(2119898)

(84)


m = 12 Array [K m] K1=120587

2 d [n

minus xminus] =

xn

n

For [s = 1 s le m2 s + +

K2s+1 =

1

2((minus1)

sd [2s + 1 120587] +s

sum

p=1

K2s+1minus2p


(2p))

K2s = (minus1)

sd [2s120587

2] +

s

sum

p=1

K2s+1minus2p


2)

2pminus1

(2p minus 1)

]



Consequently |1198781119898| = 119874(120587

2119898


100381610038161003816100381611987821198981003816100381610038161003816 le 2(

120587

2)

2119898minus11

(2119898 minus 1)

so 100381610038161003816100381611987821198981003816100381610038161003816 = 119874((

120587

2)

2119898minus11

(2119898 minus 1))

(85)


119903with a




119903for



infin

]=0((minus1)](119903+1)

(2] + 1)119903+1




119903(119909) 119861

119903(119909) and 119862

119903(119909) for any 119909 Their




Acknowledgments


References












119903






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2= 1205872


we obtain

minus4

120587

infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)3

+ 1198701119909 = 119863

2(119909) (0 le 119909 le 120587)

(14)


1= 1198632(1205872) and

consequently

1198702=120587

21198701minus 1198632(120587

2) =

1205872

8 (15)


(12) asinfin

sum

119899=1

cos 1198991199091198994

minus 1198794+ 11987921198632(119909) = minus

1

21198634(119909) +

120587

21198633(119909)

(0 le 119909 le 2120587)

(16)


4

120587

infin

sum

V=1

cos [(2V + 1) 119909](2V + 1)4

minus4

120587

infin

sum

V=1

1

(2V + 1)4+ 11987011198632(119909) = 119863

3(119909)

(0 le 119909 le 120587)

(17)


1198703=1

1205871198634(120587) minus

3

21198633(120587) +

120587

21198632(120587)

=1205872

41198701minus1

21198633(120587) =

1205873

24

(18)






1198702119904minus1

=2

120587 (2119904)(22119904

minus 1) 1205872119904 10038161003816100381610038161198612119904

1003816100381610038161003816

1198702119904=

2

120587 (2119904)(120587

2)

2119904+1

100381610038161003816100381611986421199041003816100381610038161003816 (119904 = 1 2 3 )

(19)





the assertion



119903 Exact values of119870119903


1 1205872 157079632679489661923132169162 120587

2

8 123370055013616982735431137493 120587

3

24 129192819501249250731151312774 5120587

4

384 126834753950524006818281683185 120587

5

240 127508201993867272192808879186 61120587

6

46080 127267232656453061325614987117 17120587

7

40320 127343712480668316338644619008 277120587

8

2064384 127317548065260581363477696719 31120587

9

725760 1273261242472487546381436665610 50521120587

10

3715891200 1273232382729394849508279710811 691120587

11

159667200 1273241945872154096771507790112 540553120587

12

392398110720 12732387471572495304117396905


1198792119904= (minus1)

119904

[1

21205871198632119904+1

(120587) minus1

21198632119904(120587)]

+

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198792119904minus2119901

(119904 = 1 2 3 )

(20)




1198792119904=22119904minus1

1205872119904

(2119904)

100381610038161003816100381611986121199041003816100381610038161003816 (119904 = 1 2 3 ) (21)


sum

119899=1

(minus1)119899sin 119899119909119899

= minus1

21198631(119909) (minus120587 lt 119909 lt 120587) (22)



1198762119904=(minus1)119904

21205871198632119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198762119904minus2119901

(119904 = 1 2 3 )

(23)



1198762119904=


) 1205872119904

(2119904)

100381610038161003816100381611986121199041003816100381610038161003816 (119904 = 1 2 3 ) (24)




4


5lt 1198703lt 1198701=120587

2

(25)


119870119903= 4120587




infin

sum

119899=1

sin 1198991199091198992119904minus1

= (minus1)119904

[1

21198632119904minus1

(119909) minus120587

21198632119904minus2

(119909)]

+

119904minus1

sum

119901=1

(minus1)119901+1

1198792119904minus2119901

1198632119901minus1

(119909)

infin

sum

119899=1

cos 1198991199091198992119904


[1

21198632119904(119909) minus

120587

21198632119904minus1

(119909)]

+

119904minus1

sum

119901=0

(minus1)119901

1198792119904minus2119901

1198632119901(119909)

(26)


and 1198790= 0 0 lt 119909 lt 2120587)


infin

sum

119899=1

sin 1198991199091198992119904+1


2 (2119904 + 1)(2120587)2119904+1

1198612119904+1

(119909

2120587)

((0 le 119909 le 2120587) (119904 = 1 2 3 ) 0 lt 119909 lt 2120587 for 119904 = 0)

infin

sum

119899=1

cos 1198991199091198992119904


2 (2119904)(2120587)2119904

1198612119904(119909

2120587)

(0 le 119909 le 2120587 119904 = 1 2 3 )

(27)



infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904minus1

=(minus1)119904

21198632119904minus1

(119909)

+

119904minus1

sum

119901=1

(minus1)119901+1

1198762119904minus2119901

1198632119901minus1

(119909)

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904


21198632119904(119909)

+

119904minus1

sum

119901=0

(minus1)119901

1198762119904minus2119901

1198632119901(119909)

(28)


and 1198760= 0 minus120587 lt 119909 lt 120587)


infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904+1


2 (2119904 + 1)(2120587)2119904+1

1198612119904+1

(119909 + 120587

2120587)

(minus120587 lt 119909 le 120587 119904 = 0 1 2 )

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904


2 (2119904)(2120587)2119904

1198612119904(119909 + 120587

2120587)

(minus120587 le 119909 le 120587 119904 = 1 2 3 )

(29)



infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904minus1

=120587(minus1)

119904minus1

4[1198632119904minus2

(119909) +

119904minus1

sum

119901=1

(minus1)119901

1198702119901minus1

1198632119904minus2119901minus1

(119909)]

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904

=120587(minus1)

119904

4[1198632119904minus1

(119909) +

119904

sum

119901=1

(minus1)119901

1198702119901minus1

1198632119904minus2119901

(119909)]

(30)


1 (0 lt 119909 lt 120587)


infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904+1

=(minus1)119904

1205872119904+1

4 (2119904)1198642119904(119909

120587)

(0 lt 119909 lt 120587 119904 = 0 1 2 )

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904

=(minus1)119904

1205872119904

4 (2119904 minus 1)1198642119904minus1

(119909

120587)

(0 le 119909 le 120587 119904 = 1 2 )

(31)


4

120587

infin

sum

V=0

(minus1)V cos [(2V + 1) 119909]

2V + 1= 1 (minus

120587

2lt 119909 lt

120587

2) (32)




sum

V=0


=120587(minus1)

119904minus1

4[1198632119904minus2

(119909) +

119904minus1

sum

119901=1

(minus1)119901

11987021199011198632119904minus2119901minus2

(119909)]

infin

sum

V=0

(minus1)V sin [(2V + 1) 119909]

(2V + 1)2119904

=120587(minus1)

119904minus1

4[1198632119904minus1

(119909) +

119904minus1

sum

119901=1

(minus1)119901

11987021199011198632119904minus2119901minus1

(119909)]

(33)


1198630(119909) = 1 (minus1205872 lt 119909 lt 1205872))


sum

V=0

(minus1)V cos [(2V + 1) 119909](2V + 1)2119904+1

=(minus1)119904

1205872119904+1

4 (2119904)1198642119904(119909

120587+1

2)

(minus120587

2lt 119909 lt

120587

2 119904 = 0 1 )

infin

sum

V=0

(minus1)V sin [(2V + 1) 119909]

(2V + 1)2119904=(minus1)119904minus1

1205872119904

4 (2119904 minus 1)1198642119904minus1

(119909

120587+1

2)

(minus120587

2le 119909 le

120587

2 119904 = 1 2 )

(34)




2119904(119904 =



lows the following



1198702119904minus2

=4

120587(minus1)

119904

[1

21198632119904minus1

(120587

2) minus

120587

21198632119904minus2

(120587

2)]

+

119904minus1

sum

119901=1

(minus1)119901+1

1198792119904minus2119901

1198632119901minus1

(120587

2)

1198702119904minus1

=2

120587(minus1)

119904

[1

21198632119904(120587) minus

120587

21198632119904minus1

(120587)]

+

119904minus1

sum

119901=1

(minus1)119901+1

1198792119904minus2119901

1198632119901(120587)

(35)

(119904 = 1 2 for 119904 = 11198630(119909) = 1 119879

0= 0)




sentations hold

1198762119904= 4119904


[1

21198632119904(120587

2) minus

120587

21198632119904minus1

(120587

2)]

+

119904minus1

sum

119901=0

(minus1)119901

1198792119904minus2119901

1198632119901(120587

2)

(36)

(119904 = 1 2 for 119904 = 11198630(119909) = 1)




1198702119904minus2

=4

120587(minus1)119904minus1

21198632119904minus1

(120587

2)

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901

1198632119901minus1

(120587

2)

1198702119904minus1

=2

120587(minus1)119904minus1

21198632119904(120587)

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901

1198632119901(120587)

(37)

where 119904 = 1 2 and for 119904 = 11198630(119909) = 1 119876

0= 0




sentations hold

1198762119904=

4119904

4119904 minus 1(minus1)119904

21198632119904(120587

2)

+

119904minus1

sum

119901=1

(minus1)119901+1

1198762119904minus2119901

1198632119901(120587

2)

(38)

(119904 = 1 2 for 119904 = 11198630(119909) = 1 119876

0= 0)


119903(119903 =




and 1198702119904minus1


1198702119904minus3

=(minus1)119904

1205871198632119904minus2

(120587) +

119904minus2

sum

119901=1

(minus1)119901

1198702119901minus1

1198632119904minus2119901minus1

(120587)

(39)


1198702119904minus1


1198632119904minus1

(120587

2) +

119904minus1

sum

119901=1

(minus1)119901

1198702119901minus1

1198632119904minus2119901

(120587

2)

(40)




119903(119903 = 1 2 3 ) different from

these inTheorem 1



1198702119904minus2

= (minus1)119904

1198632119904minus2

(120587

2) +

119904minus2

sum

119901=1

(minus1)119901

11987021199011198632119904minus2119901minus2

(120587

2)

1198702119904minus1


1198632119904minus1

(120587

2) +

119904minus1

sum

119901=1

(minus1)119901

11987021199011198632119904minus2119901minus1

(120587

2)

(41)


11987021198630(1205872) must be canceled


2119904minus1(119904 =


1198702119904+1

=(minus1)119904

1205871198632119904+2

(120587) +

119904

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198702119904minus2119901+1

(119904 = 0 1 )

(42)








119903=

4

120587(1 minus 2

minus(2119904+1)

) 120577 (2119904 + 1) 119903=2119904 119904 = 1 2 3

22minus4119904

120587(120577 (2119904

1

4)minus120577 (2119904

3

4))

=2

120587Γ (2119904)int

infin

0

1199052119904minus1

ch 119905119889119905 119903=2119904minus1 119904=1 2 3

(43)




2119904minus1

=2(minus1)119904

1205871198602119904minus1

(120587

2) +

119904minus1

sum

119901=1

2119904minus2119901


(2119901 minus 1)(120587

2)

2119901minus1

2119904=2(minus1)119904

1205871198602119904(120587

2) +

119904minus1

sum

119901=1

2119904minus2119901


(2119901)(120587

2)

2119901

(119904 = 1 2 3 0

def= 0)

(44)



infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2

= minus1

21198601(119909) (0 le 119909 le 120587)

infin

sum

119899=1

sin 1198991199091198992

= minus1198611(119909) (0 le 119909 le 2120587)

(45)





1= minus

4

1205871198611(120587

2) = minus

2

1205871198601(120587

2) = 1166243616123275 sdot sdot sdot

(46)


1in our paper [8]


sum

V=0

cos [(2V + 1) 119909](2V + 1)3

=1

21198602(119909) +

120587

42

(0 le 119909 le 120587)

(47)



2(1205872) + (1205874)

2and minus(1205874)

2= (12)119860

2(120587) +


2= minus

2

1205871198602(120587

2) = minus

1

1205871198602(120587) (48)


infin

sum

119899=1

cos 1198991199091198993

= 1198612(119909) + 119879

3(0 le 119909 le 2120587) (49)



2= minus

2

1205871198612(120587) = minus

2

1205871198602(120587

2)

= minus1

1205871198602(120587) = 1339193086109090 sdot sdot sdot

(50)



infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)4

=1

21198603(119909) +

120587

421198631(119909) (0 le 119909 le 120587)

(51)


3=2

1205871198603(120587

2) +

120587

22=2

1205871198603(120587

2) minus 119860

2(120587

2) (52)



infin

sum

119899=1

sin 1198991199091198994

= 1198613(119909) + 119879

31198631(119909) (0 le 119909 le 2120587) (53)


1198793= minus

1

1205871198613(120587) = 120205690 sdot sdot sdot (54)



= 1198613(1205872)minus (12)119861

3(120587)

Then

3=4

1205871198613(120587

2) minus

2

1205871198613(120587) =

2

1205871198603(120587

2) minus 119860

2(120587

2)

= 1259163310827165 sdot sdot sdot

(55)



minus

infin

sum

119899=1

cos 1198991199091198995

+ 1198795= 1198614(119909) + 119879

31198632(119909) (0 le 119909 le 2120587)

(56)

For 119909 = 120587 (56) gives (1205872)4= 1198614(120587)+(120587

2

2)1198793 Admit-


4=2

1205871198614(120587) minus 119861

3(120587) = 1278999378416936 sdot sdot sdot

(57)



minus

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)5

+120587

44=1

21198604(119909) +

120587

421198632(119909)

(0 le 119909 le 120587)

(58)


4=2

1205871198604(120587

2) minus

120587

41198602(120587

2) =

2

1205871198604(120587

2) +

1205872

82

=1

1205871198604(120587) +

1205872

42

(59)


minus

infin

sum

V=0

sin 1198991199091198996

+ 11987951198631(119909) = 119861

5(119909) + 119879

31198633(119909) (0 le 119909 le 2120587)

(60)


1198795=1

1205871198615(120587) +

1205872

31198793=1

1205871198615(120587) minus

120587

61198613(120587)


(61)


gives us minus(1205874)5+ (1205872)119879

5= 1198615(1205872) + (120587

3

23

3)1198793 Then


5= minus

120587

41198615(120587

2) +

2

1205871198615(120587) minus

120587

41198613(120587)

= 1271565517671139 sdot sdot sdot

(62)


minus

infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)6

+120587

441198631(119909) =

1

21198602(119909) +

120587

421198633(119909)

(0 le 119909 le 120587)

(63)


5= minus

2

1205871198605(120587

2) +

120587

24minus1205873

2332

= minus2

1205871198605(120587

2) + 119860

4(120587

2) minus

1205872

22 sdot 31198602(120587

2)

(64)




119903 119861119903

and 119862119903in (4) for a given 119909




1198602119904minus1

(119909) = 2(minus1)119904

infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904

+120587

2

119904minus1

sum

119901=1

(minus1)119901

21199011198632119904minus2119901minus1

(119909)

1198602119904(119909) = minus 2(minus1)

119904

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904+1

+120587

2

119904

sum

119901=1

(minus1)119901

21199011198632119904minus2119901

(119909)

(65)

119904 = 1 2 3 (0 le 119909 le 120587) 1198630= 1 for 119904 = 0119860

0(119909) =


infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904

=1

2

infin

sum

119899=1

sin 1198991199091198992119904

minus

infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905

(1198902119905

+ 1) sin119909

(1198902119905 + 1)2

minus 41198902119905cos2119909119889119905

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904+1

=1

2

infin

sum

119899=1

cos 1198991199091198992119904+1

minus

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904+1

=1

Γ (2119904+1)int

infin

0

1199052119904

119890119905

(1198902119905

minus 1) cos119909

(1198902119905+1)2

minus41198902119905cos2119909119889119905

(66)




1198612119904minus1

(119909)

= (minus1)119904

infin

sum

119899=1

sin 1198991199091198992119904

+

119904minus1

sum

119901=1

(minus1)119901

1198792119904minus2119901+1

1198632119901minus1

(119909)

1198612119904minus2

(119909)

= (minus1)119904

infin

sum

119899=1

cos 1198991199091198992119904minus1

+

119904minus1

sum

119901=1

(minus1)119901

1198792119904minus2119901+1

1198632119901minus2

(119909)

(67)

119904 = 1 2 3 (0 le 119909 le 2120587) 1198630= 1 for 119904 = 1119879

1119863119896(119896 = 0 1)



sum

119899=1

sin 1198991199091198992119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905 sin119909

1 minus 2119890119905 cos119909 + 1198902119905119889119905

(119909 = 0 119904 = 1 2 3 )

infin

sum

119899=1

cos 1198991199091198992119904minus1

=1


infin

0

1199052119904minus2

(119890119905 cos119909 minus 1)

1 minus 2119890119905 cos119909 + 1198902119905119889119905

(119909 = 0 119904 = 1 2 3 )

(68)



sum

119899=1

(minus1)119899cos 119899119909119899


1198761= minus ln 2

(69)



1198622119904minus1

(119909)

= (minus1)119904

infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901+1

1198632119901minus1

(119909)

1198622119904minus2

(119909)

= (minus1)119904

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904minus1

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901+1

1198632119901minus2

(119909)

(70)



infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

=minus1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905 sin119909

1 + 2119890119905 cos119909 + 1198902119905119889119905

(119904 = 1 2 3 )

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904minus1

=minus1


infin

0

1199052119904minus2

(119890119905 cos119909 + 1)

1 + 2119890119905 cos119909 + 1198902119905119889119905

(119904 = 1 2 )

(71)


sum

V=0

(minus1)V sin [(2V + 1) 119909]

2V + 1=minus1

2ln(tan(120587

4minus119909

2))

(minus120587

2lt 119909 lt

120587

2)

(72)




int

1205872minus119909

0

1198602119904minus1

(119906) 119889119906 = minus 2(minus1)119904

infin

sum

V=0

(minus1)V sin [(2V + 1) 119909](2V + 1)2119904+1

+120587

2

119904

sum

119901=1

(minus1)119901

21199011198632119904minus2119901

(120587

2minus 119909)

int

1205872minus119909

0

1198602119904minus2

(119906) 119889119906 = 2(minus1)119904

infin

sum

V=0

(minus1)V cos [(2V + 1) 119909]

(2V + 1)2119904

minus120587

2

119904minus1

sum

119901=1

(minus1)119901

21199011198632119904minus2119901minus1

(120587

2minus 119909)

(73)

119904 = 1 2 3 (minus1205872 le 119909 le 1205872) 1198630(119909) = 1 for 119904 = 1


infin

sum

V=0

(minus1)V sin [(2V + 1) 119909](2V + 1)2119904+1

=1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905

(1198902119905

minus 1) sin119909

(1198902119905 + 1)2

minus 41198902119905sin2119909119889119905

(119904 = 1 2 )

infin

sum

V=0

(minus1)V cos [(2V + 1) 119909]

(2V + 1)2119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905

(1198902119905

+ 1) cos119909

(1198902119905 + 1)2

minus 41198902119905sin2119909119889119905

(119904 = 1 2 )

(74)



119903(119909)minus119862

119903(119909) (0 le


inequalities [1]


4


6lt 4lt 2lt120587

2

(75)


2119904+1(119904 =



1198792119904+1

=(minus1)119904

1205871198612119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198792119904minus2119901+1

(76)

119904 = 1 2 for 119904 = 1 the term (1205872

6) 1198791declines


1198792119904+1

= 120577 (2119904 + 1) =1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905 minus 1119889119905

=minus1

(2119904)Ψ(2119904)

(1) (119904 = 1 2 )

(77)



1198762119904+1

=(minus1)119904

1205871198622119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198762119904minus2119901+1

(78)

119904 = 1 2 for 119904 = 1 the term (1205872

6) 1198761declines


1198762119904+1

= (2minus2119904

minus 1) 120577 (2119904 + 1) =minus1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905 + 1119889119905

(119904 = 1 2 )

(79)




2119904+1minus 1198762119904+1

= (1205872) 2119904(119904 = 1 2


2119904=2(minus1)119904

12058721198602119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)2119904minus2119901

(119904 = 1 2 )

(80)



1198602119904+1

(120587) = (minus1)119904

119904

sum

119901=1

(minus1)119901+1

1205872119901

2 (2119901 minus 1)2119904minus2119901+2

(119904 = 1 2 )

(81)









119903(119909) and 119862


119903 119860119903(1205872) 119861

119903(1205872) 119862

119903(1205872)


119903(120587) 119861

119903(120587) 119862

119903(120587)



1198781119898

=

119904

sum

119895=119898


120587119895119901

(2119895)1198702119904minus2119895+1

1198782119898

=

119904

sum

119895=119898


(2119895 minus 1)(120587

2)

2119895minus1

1198702119904minus2119895+1

(82)

respectively


100381610038161003816100381611987811198981003816100381610038161003816 = 119874(

1205872119898

(2119898))

100381610038161003816100381611987821198981003816100381610038161003816 = 119874((

120587

2)

2119898minus11

(2119898 minus 1))

(83)



100381610038161003816100381611987811198981003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119904

sum

119895=119898


120587119895119901

(2119895)1198702119904minus2119895+1

10038161003816100381610038161003816100381610038161003816100381610038161003816

le120587

2

119904

sum

119895=119898

1205872119895

(2119895)=120587

2

1205872119898

(2119898)

times (1 +1205872

(2119898 + 1) (2119898 + 2)+ sdot sdot sdot +

1205872119904minus2119898

(2119898 + 1) sdot sdot sdot (2119904))

le1205872119898+1

2 (2119898)(1 +

10

56+102


le1205872119898+1

2 (2119898)(1 +

1

3+1


=1205872119898+1

2 (2119898)

3

2le31205872119898

(2119898)

(84)


m = 12 Array [K m] K1=120587

2 d [n

minus xminus] =

xn

n

For [s = 1 s le m2 s + +

K2s+1 =

1

2((minus1)

sd [2s + 1 120587] +s

sum

p=1

K2s+1minus2p


(2p))

K2s = (minus1)

sd [2s120587

2] +

s

sum

p=1

K2s+1minus2p


2)

2pminus1

(2p minus 1)

]



Consequently |1198781119898| = 119874(120587

2119898


100381610038161003816100381611987821198981003816100381610038161003816 le 2(

120587

2)

2119898minus11

(2119898 minus 1)

so 100381610038161003816100381611987821198981003816100381610038161003816 = 119874((

120587

2)

2119898minus11

(2119898 minus 1))

(85)


119903with a




119903for



infin

]=0((minus1)](119903+1)

(2] + 1)119903+1




119903(119909) 119861

119903(119909) and 119862

119903(119909) for any 119909 Their




Acknowledgments


References












119903






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












infin

sum

119899=1

sin 1198991199091198992119904minus1

= (minus1)119904

[1

21198632119904minus1

(119909) minus120587

21198632119904minus2

(119909)]

+

119904minus1

sum

119901=1

(minus1)119901+1

1198792119904minus2119901

1198632119901minus1

(119909)

infin

sum

119899=1

cos 1198991199091198992119904


[1

21198632119904(119909) minus

120587

21198632119904minus1

(119909)]

+

119904minus1

sum

119901=0

(minus1)119901

1198792119904minus2119901

1198632119901(119909)

(26)


and 1198790= 0 0 lt 119909 lt 2120587)


infin

sum

119899=1

sin 1198991199091198992119904+1


2 (2119904 + 1)(2120587)2119904+1

1198612119904+1

(119909

2120587)

((0 le 119909 le 2120587) (119904 = 1 2 3 ) 0 lt 119909 lt 2120587 for 119904 = 0)

infin

sum

119899=1

cos 1198991199091198992119904


2 (2119904)(2120587)2119904

1198612119904(119909

2120587)

(0 le 119909 le 2120587 119904 = 1 2 3 )

(27)



infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904minus1

=(minus1)119904

21198632119904minus1

(119909)

+

119904minus1

sum

119901=1

(minus1)119901+1

1198762119904minus2119901

1198632119901minus1

(119909)

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904


21198632119904(119909)

+

119904minus1

sum

119901=0

(minus1)119901

1198762119904minus2119901

1198632119901(119909)

(28)


and 1198760= 0 minus120587 lt 119909 lt 120587)


infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904+1


2 (2119904 + 1)(2120587)2119904+1

1198612119904+1

(119909 + 120587

2120587)

(minus120587 lt 119909 le 120587 119904 = 0 1 2 )

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904


2 (2119904)(2120587)2119904

1198612119904(119909 + 120587

2120587)

(minus120587 le 119909 le 120587 119904 = 1 2 3 )

(29)



infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904minus1

=120587(minus1)

119904minus1

4[1198632119904minus2

(119909) +

119904minus1

sum

119901=1

(minus1)119901

1198702119901minus1

1198632119904minus2119901minus1

(119909)]

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904

=120587(minus1)

119904

4[1198632119904minus1

(119909) +

119904

sum

119901=1

(minus1)119901

1198702119901minus1

1198632119904minus2119901

(119909)]

(30)


1 (0 lt 119909 lt 120587)


infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904+1

=(minus1)119904

1205872119904+1

4 (2119904)1198642119904(119909

120587)

(0 lt 119909 lt 120587 119904 = 0 1 2 )

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904

=(minus1)119904

1205872119904

4 (2119904 minus 1)1198642119904minus1

(119909

120587)

(0 le 119909 le 120587 119904 = 1 2 )

(31)


4

120587

infin

sum

V=0

(minus1)V cos [(2V + 1) 119909]

2V + 1= 1 (minus

120587

2lt 119909 lt

120587

2) (32)




sum

V=0


=120587(minus1)

119904minus1

4[1198632119904minus2

(119909) +

119904minus1

sum

119901=1

(minus1)119901

11987021199011198632119904minus2119901minus2

(119909)]

infin

sum

V=0

(minus1)V sin [(2V + 1) 119909]

(2V + 1)2119904

=120587(minus1)

119904minus1

4[1198632119904minus1

(119909) +

119904minus1

sum

119901=1

(minus1)119901

11987021199011198632119904minus2119901minus1

(119909)]

(33)


1198630(119909) = 1 (minus1205872 lt 119909 lt 1205872))


sum

V=0

(minus1)V cos [(2V + 1) 119909](2V + 1)2119904+1

=(minus1)119904

1205872119904+1

4 (2119904)1198642119904(119909

120587+1

2)

(minus120587

2lt 119909 lt

120587

2 119904 = 0 1 )

infin

sum

V=0

(minus1)V sin [(2V + 1) 119909]

(2V + 1)2119904=(minus1)119904minus1

1205872119904

4 (2119904 minus 1)1198642119904minus1

(119909

120587+1

2)

(minus120587

2le 119909 le

120587

2 119904 = 1 2 )

(34)




2119904(119904 =



lows the following



1198702119904minus2

=4

120587(minus1)

119904

[1

21198632119904minus1

(120587

2) minus

120587

21198632119904minus2

(120587

2)]

+

119904minus1

sum

119901=1

(minus1)119901+1

1198792119904minus2119901

1198632119901minus1

(120587

2)

1198702119904minus1

=2

120587(minus1)

119904

[1

21198632119904(120587) minus

120587

21198632119904minus1

(120587)]

+

119904minus1

sum

119901=1

(minus1)119901+1

1198792119904minus2119901

1198632119901(120587)

(35)

(119904 = 1 2 for 119904 = 11198630(119909) = 1 119879

0= 0)




sentations hold

1198762119904= 4119904


[1

21198632119904(120587

2) minus

120587

21198632119904minus1

(120587

2)]

+

119904minus1

sum

119901=0

(minus1)119901

1198792119904minus2119901

1198632119901(120587

2)

(36)

(119904 = 1 2 for 119904 = 11198630(119909) = 1)




1198702119904minus2

=4

120587(minus1)119904minus1

21198632119904minus1

(120587

2)

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901

1198632119901minus1

(120587

2)

1198702119904minus1

=2

120587(minus1)119904minus1

21198632119904(120587)

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901

1198632119901(120587)

(37)

where 119904 = 1 2 and for 119904 = 11198630(119909) = 1 119876

0= 0




sentations hold

1198762119904=

4119904

4119904 minus 1(minus1)119904

21198632119904(120587

2)

+

119904minus1

sum

119901=1

(minus1)119901+1

1198762119904minus2119901

1198632119901(120587

2)

(38)

(119904 = 1 2 for 119904 = 11198630(119909) = 1 119876

0= 0)


119903(119903 =




and 1198702119904minus1


1198702119904minus3

=(minus1)119904

1205871198632119904minus2

(120587) +

119904minus2

sum

119901=1

(minus1)119901

1198702119901minus1

1198632119904minus2119901minus1

(120587)

(39)


1198702119904minus1


1198632119904minus1

(120587

2) +

119904minus1

sum

119901=1

(minus1)119901

1198702119901minus1

1198632119904minus2119901

(120587

2)

(40)




119903(119903 = 1 2 3 ) different from

these inTheorem 1



1198702119904minus2

= (minus1)119904

1198632119904minus2

(120587

2) +

119904minus2

sum

119901=1

(minus1)119901

11987021199011198632119904minus2119901minus2

(120587

2)

1198702119904minus1


1198632119904minus1

(120587

2) +

119904minus1

sum

119901=1

(minus1)119901

11987021199011198632119904minus2119901minus1

(120587

2)

(41)


11987021198630(1205872) must be canceled


2119904minus1(119904 =


1198702119904+1

=(minus1)119904

1205871198632119904+2

(120587) +

119904

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198702119904minus2119901+1

(119904 = 0 1 )

(42)








119903=

4

120587(1 minus 2

minus(2119904+1)

) 120577 (2119904 + 1) 119903=2119904 119904 = 1 2 3

22minus4119904

120587(120577 (2119904

1

4)minus120577 (2119904

3

4))

=2

120587Γ (2119904)int

infin

0

1199052119904minus1

ch 119905119889119905 119903=2119904minus1 119904=1 2 3

(43)




2119904minus1

=2(minus1)119904

1205871198602119904minus1

(120587

2) +

119904minus1

sum

119901=1

2119904minus2119901


(2119901 minus 1)(120587

2)

2119901minus1

2119904=2(minus1)119904

1205871198602119904(120587

2) +

119904minus1

sum

119901=1

2119904minus2119901


(2119901)(120587

2)

2119901

(119904 = 1 2 3 0

def= 0)

(44)



infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2

= minus1

21198601(119909) (0 le 119909 le 120587)

infin

sum

119899=1

sin 1198991199091198992

= minus1198611(119909) (0 le 119909 le 2120587)

(45)





1= minus

4

1205871198611(120587

2) = minus

2

1205871198601(120587

2) = 1166243616123275 sdot sdot sdot

(46)


1in our paper [8]


sum

V=0

cos [(2V + 1) 119909](2V + 1)3

=1

21198602(119909) +

120587

42

(0 le 119909 le 120587)

(47)



2(1205872) + (1205874)

2and minus(1205874)

2= (12)119860

2(120587) +


2= minus

2

1205871198602(120587

2) = minus

1

1205871198602(120587) (48)


infin

sum

119899=1

cos 1198991199091198993

= 1198612(119909) + 119879

3(0 le 119909 le 2120587) (49)



2= minus

2

1205871198612(120587) = minus

2

1205871198602(120587

2)

= minus1

1205871198602(120587) = 1339193086109090 sdot sdot sdot

(50)



infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)4

=1

21198603(119909) +

120587

421198631(119909) (0 le 119909 le 120587)

(51)


3=2

1205871198603(120587

2) +

120587

22=2

1205871198603(120587

2) minus 119860

2(120587

2) (52)



infin

sum

119899=1

sin 1198991199091198994

= 1198613(119909) + 119879

31198631(119909) (0 le 119909 le 2120587) (53)


1198793= minus

1

1205871198613(120587) = 120205690 sdot sdot sdot (54)



= 1198613(1205872)minus (12)119861

3(120587)

Then

3=4

1205871198613(120587

2) minus

2

1205871198613(120587) =

2

1205871198603(120587

2) minus 119860

2(120587

2)

= 1259163310827165 sdot sdot sdot

(55)



minus

infin

sum

119899=1

cos 1198991199091198995

+ 1198795= 1198614(119909) + 119879

31198632(119909) (0 le 119909 le 2120587)

(56)

For 119909 = 120587 (56) gives (1205872)4= 1198614(120587)+(120587

2

2)1198793 Admit-


4=2

1205871198614(120587) minus 119861

3(120587) = 1278999378416936 sdot sdot sdot

(57)



minus

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)5

+120587

44=1

21198604(119909) +

120587

421198632(119909)

(0 le 119909 le 120587)

(58)


4=2

1205871198604(120587

2) minus

120587

41198602(120587

2) =

2

1205871198604(120587

2) +

1205872

82

=1

1205871198604(120587) +

1205872

42

(59)


minus

infin

sum

V=0

sin 1198991199091198996

+ 11987951198631(119909) = 119861

5(119909) + 119879

31198633(119909) (0 le 119909 le 2120587)

(60)


1198795=1

1205871198615(120587) +

1205872

31198793=1

1205871198615(120587) minus

120587

61198613(120587)


(61)


gives us minus(1205874)5+ (1205872)119879

5= 1198615(1205872) + (120587

3

23

3)1198793 Then


5= minus

120587

41198615(120587

2) +

2

1205871198615(120587) minus

120587

41198613(120587)

= 1271565517671139 sdot sdot sdot

(62)


minus

infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)6

+120587

441198631(119909) =

1

21198602(119909) +

120587

421198633(119909)

(0 le 119909 le 120587)

(63)


5= minus

2

1205871198605(120587

2) +

120587

24minus1205873

2332

= minus2

1205871198605(120587

2) + 119860

4(120587

2) minus

1205872

22 sdot 31198602(120587

2)

(64)




119903 119861119903

and 119862119903in (4) for a given 119909




1198602119904minus1

(119909) = 2(minus1)119904

infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904

+120587

2

119904minus1

sum

119901=1

(minus1)119901

21199011198632119904minus2119901minus1

(119909)

1198602119904(119909) = minus 2(minus1)

119904

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904+1

+120587

2

119904

sum

119901=1

(minus1)119901

21199011198632119904minus2119901

(119909)

(65)

119904 = 1 2 3 (0 le 119909 le 120587) 1198630= 1 for 119904 = 0119860

0(119909) =


infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904

=1

2

infin

sum

119899=1

sin 1198991199091198992119904

minus

infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905

(1198902119905

+ 1) sin119909

(1198902119905 + 1)2

minus 41198902119905cos2119909119889119905

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904+1

=1

2

infin

sum

119899=1

cos 1198991199091198992119904+1

minus

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904+1

=1

Γ (2119904+1)int

infin

0

1199052119904

119890119905

(1198902119905

minus 1) cos119909

(1198902119905+1)2

minus41198902119905cos2119909119889119905

(66)




1198612119904minus1

(119909)

= (minus1)119904

infin

sum

119899=1

sin 1198991199091198992119904

+

119904minus1

sum

119901=1

(minus1)119901

1198792119904minus2119901+1

1198632119901minus1

(119909)

1198612119904minus2

(119909)

= (minus1)119904

infin

sum

119899=1

cos 1198991199091198992119904minus1

+

119904minus1

sum

119901=1

(minus1)119901

1198792119904minus2119901+1

1198632119901minus2

(119909)

(67)

119904 = 1 2 3 (0 le 119909 le 2120587) 1198630= 1 for 119904 = 1119879

1119863119896(119896 = 0 1)



sum

119899=1

sin 1198991199091198992119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905 sin119909

1 minus 2119890119905 cos119909 + 1198902119905119889119905

(119909 = 0 119904 = 1 2 3 )

infin

sum

119899=1

cos 1198991199091198992119904minus1

=1


infin

0

1199052119904minus2

(119890119905 cos119909 minus 1)

1 minus 2119890119905 cos119909 + 1198902119905119889119905

(119909 = 0 119904 = 1 2 3 )

(68)



sum

119899=1

(minus1)119899cos 119899119909119899


1198761= minus ln 2

(69)



1198622119904minus1

(119909)

= (minus1)119904

infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901+1

1198632119901minus1

(119909)

1198622119904minus2

(119909)

= (minus1)119904

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904minus1

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901+1

1198632119901minus2

(119909)

(70)



infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

=minus1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905 sin119909

1 + 2119890119905 cos119909 + 1198902119905119889119905

(119904 = 1 2 3 )

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904minus1

=minus1


infin

0

1199052119904minus2

(119890119905 cos119909 + 1)

1 + 2119890119905 cos119909 + 1198902119905119889119905

(119904 = 1 2 )

(71)


sum

V=0

(minus1)V sin [(2V + 1) 119909]

2V + 1=minus1

2ln(tan(120587

4minus119909

2))

(minus120587

2lt 119909 lt

120587

2)

(72)




int

1205872minus119909

0

1198602119904minus1

(119906) 119889119906 = minus 2(minus1)119904

infin

sum

V=0

(minus1)V sin [(2V + 1) 119909](2V + 1)2119904+1

+120587

2

119904

sum

119901=1

(minus1)119901

21199011198632119904minus2119901

(120587

2minus 119909)

int

1205872minus119909

0

1198602119904minus2

(119906) 119889119906 = 2(minus1)119904

infin

sum

V=0

(minus1)V cos [(2V + 1) 119909]

(2V + 1)2119904

minus120587

2

119904minus1

sum

119901=1

(minus1)119901

21199011198632119904minus2119901minus1

(120587

2minus 119909)

(73)

119904 = 1 2 3 (minus1205872 le 119909 le 1205872) 1198630(119909) = 1 for 119904 = 1


infin

sum

V=0

(minus1)V sin [(2V + 1) 119909](2V + 1)2119904+1

=1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905

(1198902119905

minus 1) sin119909

(1198902119905 + 1)2

minus 41198902119905sin2119909119889119905

(119904 = 1 2 )

infin

sum

V=0

(minus1)V cos [(2V + 1) 119909]

(2V + 1)2119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905

(1198902119905

+ 1) cos119909

(1198902119905 + 1)2

minus 41198902119905sin2119909119889119905

(119904 = 1 2 )

(74)



119903(119909)minus119862

119903(119909) (0 le


inequalities [1]


4


6lt 4lt 2lt120587

2

(75)


2119904+1(119904 =



1198792119904+1

=(minus1)119904

1205871198612119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198792119904minus2119901+1

(76)

119904 = 1 2 for 119904 = 1 the term (1205872

6) 1198791declines


1198792119904+1

= 120577 (2119904 + 1) =1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905 minus 1119889119905

=minus1

(2119904)Ψ(2119904)

(1) (119904 = 1 2 )

(77)



1198762119904+1

=(minus1)119904

1205871198622119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198762119904minus2119901+1

(78)

119904 = 1 2 for 119904 = 1 the term (1205872

6) 1198761declines


1198762119904+1

= (2minus2119904

minus 1) 120577 (2119904 + 1) =minus1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905 + 1119889119905

(119904 = 1 2 )

(79)




2119904+1minus 1198762119904+1

= (1205872) 2119904(119904 = 1 2


2119904=2(minus1)119904

12058721198602119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)2119904minus2119901

(119904 = 1 2 )

(80)



1198602119904+1

(120587) = (minus1)119904

119904

sum

119901=1

(minus1)119901+1

1205872119901

2 (2119901 minus 1)2119904minus2119901+2

(119904 = 1 2 )

(81)









119903(119909) and 119862


119903 119860119903(1205872) 119861

119903(1205872) 119862

119903(1205872)


119903(120587) 119861

119903(120587) 119862

119903(120587)



1198781119898

=

119904

sum

119895=119898


120587119895119901

(2119895)1198702119904minus2119895+1

1198782119898

=

119904

sum

119895=119898


(2119895 minus 1)(120587

2)

2119895minus1

1198702119904minus2119895+1

(82)

respectively


100381610038161003816100381611987811198981003816100381610038161003816 = 119874(

1205872119898

(2119898))

100381610038161003816100381611987821198981003816100381610038161003816 = 119874((

120587

2)

2119898minus11

(2119898 minus 1))

(83)



100381610038161003816100381611987811198981003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119904

sum

119895=119898


120587119895119901

(2119895)1198702119904minus2119895+1

10038161003816100381610038161003816100381610038161003816100381610038161003816

le120587

2

119904

sum

119895=119898

1205872119895

(2119895)=120587

2

1205872119898

(2119898)

times (1 +1205872

(2119898 + 1) (2119898 + 2)+ sdot sdot sdot +

1205872119904minus2119898

(2119898 + 1) sdot sdot sdot (2119904))

le1205872119898+1

2 (2119898)(1 +

10

56+102


le1205872119898+1

2 (2119898)(1 +

1

3+1


=1205872119898+1

2 (2119898)

3

2le31205872119898

(2119898)

(84)


m = 12 Array [K m] K1=120587

2 d [n

minus xminus] =

xn

n

For [s = 1 s le m2 s + +

K2s+1 =

1

2((minus1)

sd [2s + 1 120587] +s

sum

p=1

K2s+1minus2p


(2p))

K2s = (minus1)

sd [2s120587

2] +

s

sum

p=1

K2s+1minus2p


2)

2pminus1

(2p minus 1)

]



Consequently |1198781119898| = 119874(120587

2119898


100381610038161003816100381611987821198981003816100381610038161003816 le 2(

120587

2)

2119898minus11

(2119898 minus 1)

so 100381610038161003816100381611987821198981003816100381610038161003816 = 119874((

120587

2)

2119898minus11

(2119898 minus 1))

(85)


119903with a




119903for



infin

]=0((minus1)](119903+1)

(2] + 1)119903+1




119903(119909) 119861

119903(119909) and 119862

119903(119909) for any 119909 Their




Acknowledgments


References












119903






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sum

V=0


=120587(minus1)

119904minus1

4[1198632119904minus2

(119909) +

119904minus1

sum

119901=1

(minus1)119901

11987021199011198632119904minus2119901minus2

(119909)]

infin

sum

V=0

(minus1)V sin [(2V + 1) 119909]

(2V + 1)2119904

=120587(minus1)

119904minus1

4[1198632119904minus1

(119909) +

119904minus1

sum

119901=1

(minus1)119901

11987021199011198632119904minus2119901minus1

(119909)]

(33)


1198630(119909) = 1 (minus1205872 lt 119909 lt 1205872))


sum

V=0

(minus1)V cos [(2V + 1) 119909](2V + 1)2119904+1

=(minus1)119904

1205872119904+1

4 (2119904)1198642119904(119909

120587+1

2)

(minus120587

2lt 119909 lt

120587

2 119904 = 0 1 )

infin

sum

V=0

(minus1)V sin [(2V + 1) 119909]

(2V + 1)2119904=(minus1)119904minus1

1205872119904

4 (2119904 minus 1)1198642119904minus1

(119909

120587+1

2)

(minus120587

2le 119909 le

120587

2 119904 = 1 2 )

(34)




2119904(119904 =



lows the following



1198702119904minus2

=4

120587(minus1)

119904

[1

21198632119904minus1

(120587

2) minus

120587

21198632119904minus2

(120587

2)]

+

119904minus1

sum

119901=1

(minus1)119901+1

1198792119904minus2119901

1198632119901minus1

(120587

2)

1198702119904minus1

=2

120587(minus1)

119904

[1

21198632119904(120587) minus

120587

21198632119904minus1

(120587)]

+

119904minus1

sum

119901=1

(minus1)119901+1

1198792119904minus2119901

1198632119901(120587)

(35)

(119904 = 1 2 for 119904 = 11198630(119909) = 1 119879

0= 0)




sentations hold

1198762119904= 4119904


[1

21198632119904(120587

2) minus

120587

21198632119904minus1

(120587

2)]

+

119904minus1

sum

119901=0

(minus1)119901

1198792119904minus2119901

1198632119901(120587

2)

(36)

(119904 = 1 2 for 119904 = 11198630(119909) = 1)




1198702119904minus2

=4

120587(minus1)119904minus1

21198632119904minus1

(120587

2)

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901

1198632119901minus1

(120587

2)

1198702119904minus1

=2

120587(minus1)119904minus1

21198632119904(120587)

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901

1198632119901(120587)

(37)

where 119904 = 1 2 and for 119904 = 11198630(119909) = 1 119876

0= 0




sentations hold

1198762119904=

4119904

4119904 minus 1(minus1)119904

21198632119904(120587

2)

+

119904minus1

sum

119901=1

(minus1)119901+1

1198762119904minus2119901

1198632119901(120587

2)

(38)

(119904 = 1 2 for 119904 = 11198630(119909) = 1 119876

0= 0)


119903(119903 =




and 1198702119904minus1


1198702119904minus3

=(minus1)119904

1205871198632119904minus2

(120587) +

119904minus2

sum

119901=1

(minus1)119901

1198702119901minus1

1198632119904minus2119901minus1

(120587)

(39)


1198702119904minus1


1198632119904minus1

(120587

2) +

119904minus1

sum

119901=1

(minus1)119901

1198702119901minus1

1198632119904minus2119901

(120587

2)

(40)




119903(119903 = 1 2 3 ) different from

these inTheorem 1



1198702119904minus2

= (minus1)119904

1198632119904minus2

(120587

2) +

119904minus2

sum

119901=1

(minus1)119901

11987021199011198632119904minus2119901minus2

(120587

2)

1198702119904minus1


1198632119904minus1

(120587

2) +

119904minus1

sum

119901=1

(minus1)119901

11987021199011198632119904minus2119901minus1

(120587

2)

(41)


11987021198630(1205872) must be canceled


2119904minus1(119904 =


1198702119904+1

=(minus1)119904

1205871198632119904+2

(120587) +

119904

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198702119904minus2119901+1

(119904 = 0 1 )

(42)








119903=

4

120587(1 minus 2

minus(2119904+1)

) 120577 (2119904 + 1) 119903=2119904 119904 = 1 2 3

22minus4119904

120587(120577 (2119904

1

4)minus120577 (2119904

3

4))

=2

120587Γ (2119904)int

infin

0

1199052119904minus1

ch 119905119889119905 119903=2119904minus1 119904=1 2 3

(43)




2119904minus1

=2(minus1)119904

1205871198602119904minus1

(120587

2) +

119904minus1

sum

119901=1

2119904minus2119901


(2119901 minus 1)(120587

2)

2119901minus1

2119904=2(minus1)119904

1205871198602119904(120587

2) +

119904minus1

sum

119901=1

2119904minus2119901


(2119901)(120587

2)

2119901

(119904 = 1 2 3 0

def= 0)

(44)



infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2

= minus1

21198601(119909) (0 le 119909 le 120587)

infin

sum

119899=1

sin 1198991199091198992

= minus1198611(119909) (0 le 119909 le 2120587)

(45)





1= minus

4

1205871198611(120587

2) = minus

2

1205871198601(120587

2) = 1166243616123275 sdot sdot sdot

(46)


1in our paper [8]


sum

V=0

cos [(2V + 1) 119909](2V + 1)3

=1

21198602(119909) +

120587

42

(0 le 119909 le 120587)

(47)



2(1205872) + (1205874)

2and minus(1205874)

2= (12)119860

2(120587) +


2= minus

2

1205871198602(120587

2) = minus

1

1205871198602(120587) (48)


infin

sum

119899=1

cos 1198991199091198993

= 1198612(119909) + 119879

3(0 le 119909 le 2120587) (49)



2= minus

2

1205871198612(120587) = minus

2

1205871198602(120587

2)

= minus1

1205871198602(120587) = 1339193086109090 sdot sdot sdot

(50)



infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)4

=1

21198603(119909) +

120587

421198631(119909) (0 le 119909 le 120587)

(51)


3=2

1205871198603(120587

2) +

120587

22=2

1205871198603(120587

2) minus 119860

2(120587

2) (52)



infin

sum

119899=1

sin 1198991199091198994

= 1198613(119909) + 119879

31198631(119909) (0 le 119909 le 2120587) (53)


1198793= minus

1

1205871198613(120587) = 120205690 sdot sdot sdot (54)



= 1198613(1205872)minus (12)119861

3(120587)

Then

3=4

1205871198613(120587

2) minus

2

1205871198613(120587) =

2

1205871198603(120587

2) minus 119860

2(120587

2)

= 1259163310827165 sdot sdot sdot

(55)



minus

infin

sum

119899=1

cos 1198991199091198995

+ 1198795= 1198614(119909) + 119879

31198632(119909) (0 le 119909 le 2120587)

(56)

For 119909 = 120587 (56) gives (1205872)4= 1198614(120587)+(120587

2

2)1198793 Admit-


4=2

1205871198614(120587) minus 119861

3(120587) = 1278999378416936 sdot sdot sdot

(57)



minus

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)5

+120587

44=1

21198604(119909) +

120587

421198632(119909)

(0 le 119909 le 120587)

(58)


4=2

1205871198604(120587

2) minus

120587

41198602(120587

2) =

2

1205871198604(120587

2) +

1205872

82

=1

1205871198604(120587) +

1205872

42

(59)


minus

infin

sum

V=0

sin 1198991199091198996

+ 11987951198631(119909) = 119861

5(119909) + 119879

31198633(119909) (0 le 119909 le 2120587)

(60)


1198795=1

1205871198615(120587) +

1205872

31198793=1

1205871198615(120587) minus

120587

61198613(120587)


(61)


gives us minus(1205874)5+ (1205872)119879

5= 1198615(1205872) + (120587

3

23

3)1198793 Then


5= minus

120587

41198615(120587

2) +

2

1205871198615(120587) minus

120587

41198613(120587)

= 1271565517671139 sdot sdot sdot

(62)


minus

infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)6

+120587

441198631(119909) =

1

21198602(119909) +

120587

421198633(119909)

(0 le 119909 le 120587)

(63)


5= minus

2

1205871198605(120587

2) +

120587

24minus1205873

2332

= minus2

1205871198605(120587

2) + 119860

4(120587

2) minus

1205872

22 sdot 31198602(120587

2)

(64)




119903 119861119903

and 119862119903in (4) for a given 119909




1198602119904minus1

(119909) = 2(minus1)119904

infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904

+120587

2

119904minus1

sum

119901=1

(minus1)119901

21199011198632119904minus2119901minus1

(119909)

1198602119904(119909) = minus 2(minus1)

119904

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904+1

+120587

2

119904

sum

119901=1

(minus1)119901

21199011198632119904minus2119901

(119909)

(65)

119904 = 1 2 3 (0 le 119909 le 120587) 1198630= 1 for 119904 = 0119860

0(119909) =


infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904

=1

2

infin

sum

119899=1

sin 1198991199091198992119904

minus

infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905

(1198902119905

+ 1) sin119909

(1198902119905 + 1)2

minus 41198902119905cos2119909119889119905

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904+1

=1

2

infin

sum

119899=1

cos 1198991199091198992119904+1

minus

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904+1

=1

Γ (2119904+1)int

infin

0

1199052119904

119890119905

(1198902119905

minus 1) cos119909

(1198902119905+1)2

minus41198902119905cos2119909119889119905

(66)




1198612119904minus1

(119909)

= (minus1)119904

infin

sum

119899=1

sin 1198991199091198992119904

+

119904minus1

sum

119901=1

(minus1)119901

1198792119904minus2119901+1

1198632119901minus1

(119909)

1198612119904minus2

(119909)

= (minus1)119904

infin

sum

119899=1

cos 1198991199091198992119904minus1

+

119904minus1

sum

119901=1

(minus1)119901

1198792119904minus2119901+1

1198632119901minus2

(119909)

(67)

119904 = 1 2 3 (0 le 119909 le 2120587) 1198630= 1 for 119904 = 1119879

1119863119896(119896 = 0 1)



sum

119899=1

sin 1198991199091198992119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905 sin119909

1 minus 2119890119905 cos119909 + 1198902119905119889119905

(119909 = 0 119904 = 1 2 3 )

infin

sum

119899=1

cos 1198991199091198992119904minus1

=1


infin

0

1199052119904minus2

(119890119905 cos119909 minus 1)

1 minus 2119890119905 cos119909 + 1198902119905119889119905

(119909 = 0 119904 = 1 2 3 )

(68)



sum

119899=1

(minus1)119899cos 119899119909119899


1198761= minus ln 2

(69)



1198622119904minus1

(119909)

= (minus1)119904

infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901+1

1198632119901minus1

(119909)

1198622119904minus2

(119909)

= (minus1)119904

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904minus1

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901+1

1198632119901minus2

(119909)

(70)



infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

=minus1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905 sin119909

1 + 2119890119905 cos119909 + 1198902119905119889119905

(119904 = 1 2 3 )

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904minus1

=minus1


infin

0

1199052119904minus2

(119890119905 cos119909 + 1)

1 + 2119890119905 cos119909 + 1198902119905119889119905

(119904 = 1 2 )

(71)


sum

V=0

(minus1)V sin [(2V + 1) 119909]

2V + 1=minus1

2ln(tan(120587

4minus119909

2))

(minus120587

2lt 119909 lt

120587

2)

(72)




int

1205872minus119909

0

1198602119904minus1

(119906) 119889119906 = minus 2(minus1)119904

infin

sum

V=0

(minus1)V sin [(2V + 1) 119909](2V + 1)2119904+1

+120587

2

119904

sum

119901=1

(minus1)119901

21199011198632119904minus2119901

(120587

2minus 119909)

int

1205872minus119909

0

1198602119904minus2

(119906) 119889119906 = 2(minus1)119904

infin

sum

V=0

(minus1)V cos [(2V + 1) 119909]

(2V + 1)2119904

minus120587

2

119904minus1

sum

119901=1

(minus1)119901

21199011198632119904minus2119901minus1

(120587

2minus 119909)

(73)

119904 = 1 2 3 (minus1205872 le 119909 le 1205872) 1198630(119909) = 1 for 119904 = 1


infin

sum

V=0

(minus1)V sin [(2V + 1) 119909](2V + 1)2119904+1

=1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905

(1198902119905

minus 1) sin119909

(1198902119905 + 1)2

minus 41198902119905sin2119909119889119905

(119904 = 1 2 )

infin

sum

V=0

(minus1)V cos [(2V + 1) 119909]

(2V + 1)2119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905

(1198902119905

+ 1) cos119909

(1198902119905 + 1)2

minus 41198902119905sin2119909119889119905

(119904 = 1 2 )

(74)



119903(119909)minus119862

119903(119909) (0 le


inequalities [1]


4


6lt 4lt 2lt120587

2

(75)


2119904+1(119904 =



1198792119904+1

=(minus1)119904

1205871198612119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198792119904minus2119901+1

(76)

119904 = 1 2 for 119904 = 1 the term (1205872

6) 1198791declines


1198792119904+1

= 120577 (2119904 + 1) =1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905 minus 1119889119905

=minus1

(2119904)Ψ(2119904)

(1) (119904 = 1 2 )

(77)



1198762119904+1

=(minus1)119904

1205871198622119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198762119904minus2119901+1

(78)

119904 = 1 2 for 119904 = 1 the term (1205872

6) 1198761declines


1198762119904+1

= (2minus2119904

minus 1) 120577 (2119904 + 1) =minus1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905 + 1119889119905

(119904 = 1 2 )

(79)




2119904+1minus 1198762119904+1

= (1205872) 2119904(119904 = 1 2


2119904=2(minus1)119904

12058721198602119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)2119904minus2119901

(119904 = 1 2 )

(80)



1198602119904+1

(120587) = (minus1)119904

119904

sum

119901=1

(minus1)119901+1

1205872119901

2 (2119901 minus 1)2119904minus2119901+2

(119904 = 1 2 )

(81)









119903(119909) and 119862


119903 119860119903(1205872) 119861

119903(1205872) 119862

119903(1205872)


119903(120587) 119861

119903(120587) 119862

119903(120587)



1198781119898

=

119904

sum

119895=119898


120587119895119901

(2119895)1198702119904minus2119895+1

1198782119898

=

119904

sum

119895=119898


(2119895 minus 1)(120587

2)

2119895minus1

1198702119904minus2119895+1

(82)

respectively


100381610038161003816100381611987811198981003816100381610038161003816 = 119874(

1205872119898

(2119898))

100381610038161003816100381611987821198981003816100381610038161003816 = 119874((

120587

2)

2119898minus11

(2119898 minus 1))

(83)



100381610038161003816100381611987811198981003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119904

sum

119895=119898


120587119895119901

(2119895)1198702119904minus2119895+1

10038161003816100381610038161003816100381610038161003816100381610038161003816

le120587

2

119904

sum

119895=119898

1205872119895

(2119895)=120587

2

1205872119898

(2119898)

times (1 +1205872

(2119898 + 1) (2119898 + 2)+ sdot sdot sdot +

1205872119904minus2119898

(2119898 + 1) sdot sdot sdot (2119904))

le1205872119898+1

2 (2119898)(1 +

10

56+102


le1205872119898+1

2 (2119898)(1 +

1

3+1


=1205872119898+1

2 (2119898)

3

2le31205872119898

(2119898)

(84)


m = 12 Array [K m] K1=120587

2 d [n

minus xminus] =

xn

n

For [s = 1 s le m2 s + +

K2s+1 =

1

2((minus1)

sd [2s + 1 120587] +s

sum

p=1

K2s+1minus2p


(2p))

K2s = (minus1)

sd [2s120587

2] +

s

sum

p=1

K2s+1minus2p


2)

2pminus1

(2p minus 1)

]



Consequently |1198781119898| = 119874(120587

2119898


100381610038161003816100381611987821198981003816100381610038161003816 le 2(

120587

2)

2119898minus11

(2119898 minus 1)

so 100381610038161003816100381611987821198981003816100381610038161003816 = 119874((

120587

2)

2119898minus11

(2119898 minus 1))

(85)


119903with a




119903for



infin

]=0((minus1)](119903+1)

(2] + 1)119903+1




119903(119909) 119861

119903(119909) and 119862

119903(119909) for any 119909 Their




Acknowledgments


References












119903






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











and 1198702119904minus1


1198702119904minus3

=(minus1)119904

1205871198632119904minus2

(120587) +

119904minus2

sum

119901=1

(minus1)119901

1198702119901minus1

1198632119904minus2119901minus1

(120587)

(39)


1198702119904minus1


1198632119904minus1

(120587

2) +

119904minus1

sum

119901=1

(minus1)119901

1198702119901minus1

1198632119904minus2119901

(120587

2)

(40)




119903(119903 = 1 2 3 ) different from

these inTheorem 1



1198702119904minus2

= (minus1)119904

1198632119904minus2

(120587

2) +

119904minus2

sum

119901=1

(minus1)119901

11987021199011198632119904minus2119901minus2

(120587

2)

1198702119904minus1


1198632119904minus1

(120587

2) +

119904minus1

sum

119901=1

(minus1)119901

11987021199011198632119904minus2119901minus1

(120587

2)

(41)


11987021198630(1205872) must be canceled


2119904minus1(119904 =


1198702119904+1

=(minus1)119904

1205871198632119904+2

(120587) +

119904

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198702119904minus2119901+1

(119904 = 0 1 )

(42)








119903=

4

120587(1 minus 2

minus(2119904+1)

) 120577 (2119904 + 1) 119903=2119904 119904 = 1 2 3

22minus4119904

120587(120577 (2119904

1

4)minus120577 (2119904

3

4))

=2

120587Γ (2119904)int

infin

0

1199052119904minus1

ch 119905119889119905 119903=2119904minus1 119904=1 2 3

(43)




2119904minus1

=2(minus1)119904

1205871198602119904minus1

(120587

2) +

119904minus1

sum

119901=1

2119904minus2119901


(2119901 minus 1)(120587

2)

2119901minus1

2119904=2(minus1)119904

1205871198602119904(120587

2) +

119904minus1

sum

119901=1

2119904minus2119901


(2119901)(120587

2)

2119901

(119904 = 1 2 3 0

def= 0)

(44)



infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2

= minus1

21198601(119909) (0 le 119909 le 120587)

infin

sum

119899=1

sin 1198991199091198992

= minus1198611(119909) (0 le 119909 le 2120587)

(45)





1= minus

4

1205871198611(120587

2) = minus

2

1205871198601(120587

2) = 1166243616123275 sdot sdot sdot

(46)


1in our paper [8]


sum

V=0

cos [(2V + 1) 119909](2V + 1)3

=1

21198602(119909) +

120587

42

(0 le 119909 le 120587)

(47)



2(1205872) + (1205874)

2and minus(1205874)

2= (12)119860

2(120587) +


2= minus

2

1205871198602(120587

2) = minus

1

1205871198602(120587) (48)


infin

sum

119899=1

cos 1198991199091198993

= 1198612(119909) + 119879

3(0 le 119909 le 2120587) (49)



2= minus

2

1205871198612(120587) = minus

2

1205871198602(120587

2)

= minus1

1205871198602(120587) = 1339193086109090 sdot sdot sdot

(50)



infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)4

=1

21198603(119909) +

120587

421198631(119909) (0 le 119909 le 120587)

(51)


3=2

1205871198603(120587

2) +

120587

22=2

1205871198603(120587

2) minus 119860

2(120587

2) (52)



infin

sum

119899=1

sin 1198991199091198994

= 1198613(119909) + 119879

31198631(119909) (0 le 119909 le 2120587) (53)


1198793= minus

1

1205871198613(120587) = 120205690 sdot sdot sdot (54)



= 1198613(1205872)minus (12)119861

3(120587)

Then

3=4

1205871198613(120587

2) minus

2

1205871198613(120587) =

2

1205871198603(120587

2) minus 119860

2(120587

2)

= 1259163310827165 sdot sdot sdot

(55)



minus

infin

sum

119899=1

cos 1198991199091198995

+ 1198795= 1198614(119909) + 119879

31198632(119909) (0 le 119909 le 2120587)

(56)

For 119909 = 120587 (56) gives (1205872)4= 1198614(120587)+(120587

2

2)1198793 Admit-


4=2

1205871198614(120587) minus 119861

3(120587) = 1278999378416936 sdot sdot sdot

(57)



minus

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)5

+120587

44=1

21198604(119909) +

120587

421198632(119909)

(0 le 119909 le 120587)

(58)


4=2

1205871198604(120587

2) minus

120587

41198602(120587

2) =

2

1205871198604(120587

2) +

1205872

82

=1

1205871198604(120587) +

1205872

42

(59)


minus

infin

sum

V=0

sin 1198991199091198996

+ 11987951198631(119909) = 119861

5(119909) + 119879

31198633(119909) (0 le 119909 le 2120587)

(60)


1198795=1

1205871198615(120587) +

1205872

31198793=1

1205871198615(120587) minus

120587

61198613(120587)


(61)


gives us minus(1205874)5+ (1205872)119879

5= 1198615(1205872) + (120587

3

23

3)1198793 Then


5= minus

120587

41198615(120587

2) +

2

1205871198615(120587) minus

120587

41198613(120587)

= 1271565517671139 sdot sdot sdot

(62)


minus

infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)6

+120587

441198631(119909) =

1

21198602(119909) +

120587

421198633(119909)

(0 le 119909 le 120587)

(63)


5= minus

2

1205871198605(120587

2) +

120587

24minus1205873

2332

= minus2

1205871198605(120587

2) + 119860

4(120587

2) minus

1205872

22 sdot 31198602(120587

2)

(64)




119903 119861119903

and 119862119903in (4) for a given 119909




1198602119904minus1

(119909) = 2(minus1)119904

infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904

+120587

2

119904minus1

sum

119901=1

(minus1)119901

21199011198632119904minus2119901minus1

(119909)

1198602119904(119909) = minus 2(minus1)

119904

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904+1

+120587

2

119904

sum

119901=1

(minus1)119901

21199011198632119904minus2119901

(119909)

(65)

119904 = 1 2 3 (0 le 119909 le 120587) 1198630= 1 for 119904 = 0119860

0(119909) =


infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904

=1

2

infin

sum

119899=1

sin 1198991199091198992119904

minus

infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905

(1198902119905

+ 1) sin119909

(1198902119905 + 1)2

minus 41198902119905cos2119909119889119905

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904+1

=1

2

infin

sum

119899=1

cos 1198991199091198992119904+1

minus

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904+1

=1

Γ (2119904+1)int

infin

0

1199052119904

119890119905

(1198902119905

minus 1) cos119909

(1198902119905+1)2

minus41198902119905cos2119909119889119905

(66)




1198612119904minus1

(119909)

= (minus1)119904

infin

sum

119899=1

sin 1198991199091198992119904

+

119904minus1

sum

119901=1

(minus1)119901

1198792119904minus2119901+1

1198632119901minus1

(119909)

1198612119904minus2

(119909)

= (minus1)119904

infin

sum

119899=1

cos 1198991199091198992119904minus1

+

119904minus1

sum

119901=1

(minus1)119901

1198792119904minus2119901+1

1198632119901minus2

(119909)

(67)

119904 = 1 2 3 (0 le 119909 le 2120587) 1198630= 1 for 119904 = 1119879

1119863119896(119896 = 0 1)



sum

119899=1

sin 1198991199091198992119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905 sin119909

1 minus 2119890119905 cos119909 + 1198902119905119889119905

(119909 = 0 119904 = 1 2 3 )

infin

sum

119899=1

cos 1198991199091198992119904minus1

=1


infin

0

1199052119904minus2

(119890119905 cos119909 minus 1)

1 minus 2119890119905 cos119909 + 1198902119905119889119905

(119909 = 0 119904 = 1 2 3 )

(68)



sum

119899=1

(minus1)119899cos 119899119909119899


1198761= minus ln 2

(69)



1198622119904minus1

(119909)

= (minus1)119904

infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901+1

1198632119901minus1

(119909)

1198622119904minus2

(119909)

= (minus1)119904

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904minus1

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901+1

1198632119901minus2

(119909)

(70)



infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

=minus1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905 sin119909

1 + 2119890119905 cos119909 + 1198902119905119889119905

(119904 = 1 2 3 )

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904minus1

=minus1


infin

0

1199052119904minus2

(119890119905 cos119909 + 1)

1 + 2119890119905 cos119909 + 1198902119905119889119905

(119904 = 1 2 )

(71)


sum

V=0

(minus1)V sin [(2V + 1) 119909]

2V + 1=minus1

2ln(tan(120587

4minus119909

2))

(minus120587

2lt 119909 lt

120587

2)

(72)




int

1205872minus119909

0

1198602119904minus1

(119906) 119889119906 = minus 2(minus1)119904

infin

sum

V=0

(minus1)V sin [(2V + 1) 119909](2V + 1)2119904+1

+120587

2

119904

sum

119901=1

(minus1)119901

21199011198632119904minus2119901

(120587

2minus 119909)

int

1205872minus119909

0

1198602119904minus2

(119906) 119889119906 = 2(minus1)119904

infin

sum

V=0

(minus1)V cos [(2V + 1) 119909]

(2V + 1)2119904

minus120587

2

119904minus1

sum

119901=1

(minus1)119901

21199011198632119904minus2119901minus1

(120587

2minus 119909)

(73)

119904 = 1 2 3 (minus1205872 le 119909 le 1205872) 1198630(119909) = 1 for 119904 = 1


infin

sum

V=0

(minus1)V sin [(2V + 1) 119909](2V + 1)2119904+1

=1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905

(1198902119905

minus 1) sin119909

(1198902119905 + 1)2

minus 41198902119905sin2119909119889119905

(119904 = 1 2 )

infin

sum

V=0

(minus1)V cos [(2V + 1) 119909]

(2V + 1)2119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905

(1198902119905

+ 1) cos119909

(1198902119905 + 1)2

minus 41198902119905sin2119909119889119905

(119904 = 1 2 )

(74)



119903(119909)minus119862

119903(119909) (0 le


inequalities [1]


4


6lt 4lt 2lt120587

2

(75)


2119904+1(119904 =



1198792119904+1

=(minus1)119904

1205871198612119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198792119904minus2119901+1

(76)

119904 = 1 2 for 119904 = 1 the term (1205872

6) 1198791declines


1198792119904+1

= 120577 (2119904 + 1) =1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905 minus 1119889119905

=minus1

(2119904)Ψ(2119904)

(1) (119904 = 1 2 )

(77)



1198762119904+1

=(minus1)119904

1205871198622119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198762119904minus2119901+1

(78)

119904 = 1 2 for 119904 = 1 the term (1205872

6) 1198761declines


1198762119904+1

= (2minus2119904

minus 1) 120577 (2119904 + 1) =minus1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905 + 1119889119905

(119904 = 1 2 )

(79)




2119904+1minus 1198762119904+1

= (1205872) 2119904(119904 = 1 2


2119904=2(minus1)119904

12058721198602119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)2119904minus2119901

(119904 = 1 2 )

(80)



1198602119904+1

(120587) = (minus1)119904

119904

sum

119901=1

(minus1)119901+1

1205872119901

2 (2119901 minus 1)2119904minus2119901+2

(119904 = 1 2 )

(81)









119903(119909) and 119862


119903 119860119903(1205872) 119861

119903(1205872) 119862

119903(1205872)


119903(120587) 119861

119903(120587) 119862

119903(120587)



1198781119898

=

119904

sum

119895=119898


120587119895119901

(2119895)1198702119904minus2119895+1

1198782119898

=

119904

sum

119895=119898


(2119895 minus 1)(120587

2)

2119895minus1

1198702119904minus2119895+1

(82)

respectively


100381610038161003816100381611987811198981003816100381610038161003816 = 119874(

1205872119898

(2119898))

100381610038161003816100381611987821198981003816100381610038161003816 = 119874((

120587

2)

2119898minus11

(2119898 minus 1))

(83)



100381610038161003816100381611987811198981003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119904

sum

119895=119898


120587119895119901

(2119895)1198702119904minus2119895+1

10038161003816100381610038161003816100381610038161003816100381610038161003816

le120587

2

119904

sum

119895=119898

1205872119895

(2119895)=120587

2

1205872119898

(2119898)

times (1 +1205872

(2119898 + 1) (2119898 + 2)+ sdot sdot sdot +

1205872119904minus2119898

(2119898 + 1) sdot sdot sdot (2119904))

le1205872119898+1

2 (2119898)(1 +

10

56+102


le1205872119898+1

2 (2119898)(1 +

1

3+1


=1205872119898+1

2 (2119898)

3

2le31205872119898

(2119898)

(84)


m = 12 Array [K m] K1=120587

2 d [n

minus xminus] =

xn

n

For [s = 1 s le m2 s + +

K2s+1 =

1

2((minus1)

sd [2s + 1 120587] +s

sum

p=1

K2s+1minus2p


(2p))

K2s = (minus1)

sd [2s120587

2] +

s

sum

p=1

K2s+1minus2p


2)

2pminus1

(2p minus 1)

]



Consequently |1198781119898| = 119874(120587

2119898


100381610038161003816100381611987821198981003816100381610038161003816 le 2(

120587

2)

2119898minus11

(2119898 minus 1)

so 100381610038161003816100381611987821198981003816100381610038161003816 = 119874((

120587

2)

2119898minus11

(2119898 minus 1))

(85)


119903with a




119903for



infin

]=0((minus1)](119903+1)

(2] + 1)119903+1




119903(119909) 119861

119903(119909) and 119862

119903(119909) for any 119909 Their




Acknowledgments


References












119903






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2(1205872) + (1205874)

2and minus(1205874)

2= (12)119860

2(120587) +


2= minus

2

1205871198602(120587

2) = minus

1

1205871198602(120587) (48)


infin

sum

119899=1

cos 1198991199091198993

= 1198612(119909) + 119879

3(0 le 119909 le 2120587) (49)



2= minus

2

1205871198612(120587) = minus

2

1205871198602(120587

2)

= minus1

1205871198602(120587) = 1339193086109090 sdot sdot sdot

(50)



infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)4

=1

21198603(119909) +

120587

421198631(119909) (0 le 119909 le 120587)

(51)


3=2

1205871198603(120587

2) +

120587

22=2

1205871198603(120587

2) minus 119860

2(120587

2) (52)



infin

sum

119899=1

sin 1198991199091198994

= 1198613(119909) + 119879

31198631(119909) (0 le 119909 le 2120587) (53)


1198793= minus

1

1205871198613(120587) = 120205690 sdot sdot sdot (54)



= 1198613(1205872)minus (12)119861

3(120587)

Then

3=4

1205871198613(120587

2) minus

2

1205871198613(120587) =

2

1205871198603(120587

2) minus 119860

2(120587

2)

= 1259163310827165 sdot sdot sdot

(55)



minus

infin

sum

119899=1

cos 1198991199091198995

+ 1198795= 1198614(119909) + 119879

31198632(119909) (0 le 119909 le 2120587)

(56)

For 119909 = 120587 (56) gives (1205872)4= 1198614(120587)+(120587

2

2)1198793 Admit-


4=2

1205871198614(120587) minus 119861

3(120587) = 1278999378416936 sdot sdot sdot

(57)



minus

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)5

+120587

44=1

21198604(119909) +

120587

421198632(119909)

(0 le 119909 le 120587)

(58)


4=2

1205871198604(120587

2) minus

120587

41198602(120587

2) =

2

1205871198604(120587

2) +

1205872

82

=1

1205871198604(120587) +

1205872

42

(59)


minus

infin

sum

V=0

sin 1198991199091198996

+ 11987951198631(119909) = 119861

5(119909) + 119879

31198633(119909) (0 le 119909 le 2120587)

(60)


1198795=1

1205871198615(120587) +

1205872

31198793=1

1205871198615(120587) minus

120587

61198613(120587)


(61)


gives us minus(1205874)5+ (1205872)119879

5= 1198615(1205872) + (120587

3

23

3)1198793 Then


5= minus

120587

41198615(120587

2) +

2

1205871198615(120587) minus

120587

41198613(120587)

= 1271565517671139 sdot sdot sdot

(62)


minus

infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)6

+120587

441198631(119909) =

1

21198602(119909) +

120587

421198633(119909)

(0 le 119909 le 120587)

(63)


5= minus

2

1205871198605(120587

2) +

120587

24minus1205873

2332

= minus2

1205871198605(120587

2) + 119860

4(120587

2) minus

1205872

22 sdot 31198602(120587

2)

(64)




119903 119861119903

and 119862119903in (4) for a given 119909




1198602119904minus1

(119909) = 2(minus1)119904

infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904

+120587

2

119904minus1

sum

119901=1

(minus1)119901

21199011198632119904minus2119901minus1

(119909)

1198602119904(119909) = minus 2(minus1)

119904

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904+1

+120587

2

119904

sum

119901=1

(minus1)119901

21199011198632119904minus2119901

(119909)

(65)

119904 = 1 2 3 (0 le 119909 le 120587) 1198630= 1 for 119904 = 0119860

0(119909) =


infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904

=1

2

infin

sum

119899=1

sin 1198991199091198992119904

minus

infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905

(1198902119905

+ 1) sin119909

(1198902119905 + 1)2

minus 41198902119905cos2119909119889119905

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904+1

=1

2

infin

sum

119899=1

cos 1198991199091198992119904+1

minus

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904+1

=1

Γ (2119904+1)int

infin

0

1199052119904

119890119905

(1198902119905

minus 1) cos119909

(1198902119905+1)2

minus41198902119905cos2119909119889119905

(66)




1198612119904minus1

(119909)

= (minus1)119904

infin

sum

119899=1

sin 1198991199091198992119904

+

119904minus1

sum

119901=1

(minus1)119901

1198792119904minus2119901+1

1198632119901minus1

(119909)

1198612119904minus2

(119909)

= (minus1)119904

infin

sum

119899=1

cos 1198991199091198992119904minus1

+

119904minus1

sum

119901=1

(minus1)119901

1198792119904minus2119901+1

1198632119901minus2

(119909)

(67)

119904 = 1 2 3 (0 le 119909 le 2120587) 1198630= 1 for 119904 = 1119879

1119863119896(119896 = 0 1)



sum

119899=1

sin 1198991199091198992119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905 sin119909

1 minus 2119890119905 cos119909 + 1198902119905119889119905

(119909 = 0 119904 = 1 2 3 )

infin

sum

119899=1

cos 1198991199091198992119904minus1

=1


infin

0

1199052119904minus2

(119890119905 cos119909 minus 1)

1 minus 2119890119905 cos119909 + 1198902119905119889119905

(119909 = 0 119904 = 1 2 3 )

(68)



sum

119899=1

(minus1)119899cos 119899119909119899


1198761= minus ln 2

(69)



1198622119904minus1

(119909)

= (minus1)119904

infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901+1

1198632119901minus1

(119909)

1198622119904minus2

(119909)

= (minus1)119904

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904minus1

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901+1

1198632119901minus2

(119909)

(70)



infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

=minus1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905 sin119909

1 + 2119890119905 cos119909 + 1198902119905119889119905

(119904 = 1 2 3 )

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904minus1

=minus1


infin

0

1199052119904minus2

(119890119905 cos119909 + 1)

1 + 2119890119905 cos119909 + 1198902119905119889119905

(119904 = 1 2 )

(71)


sum

V=0

(minus1)V sin [(2V + 1) 119909]

2V + 1=minus1

2ln(tan(120587

4minus119909

2))

(minus120587

2lt 119909 lt

120587

2)

(72)




int

1205872minus119909

0

1198602119904minus1

(119906) 119889119906 = minus 2(minus1)119904

infin

sum

V=0

(minus1)V sin [(2V + 1) 119909](2V + 1)2119904+1

+120587

2

119904

sum

119901=1

(minus1)119901

21199011198632119904minus2119901

(120587

2minus 119909)

int

1205872minus119909

0

1198602119904minus2

(119906) 119889119906 = 2(minus1)119904

infin

sum

V=0

(minus1)V cos [(2V + 1) 119909]

(2V + 1)2119904

minus120587

2

119904minus1

sum

119901=1

(minus1)119901

21199011198632119904minus2119901minus1

(120587

2minus 119909)

(73)

119904 = 1 2 3 (minus1205872 le 119909 le 1205872) 1198630(119909) = 1 for 119904 = 1


infin

sum

V=0

(minus1)V sin [(2V + 1) 119909](2V + 1)2119904+1

=1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905

(1198902119905

minus 1) sin119909

(1198902119905 + 1)2

minus 41198902119905sin2119909119889119905

(119904 = 1 2 )

infin

sum

V=0

(minus1)V cos [(2V + 1) 119909]

(2V + 1)2119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905

(1198902119905

+ 1) cos119909

(1198902119905 + 1)2

minus 41198902119905sin2119909119889119905

(119904 = 1 2 )

(74)



119903(119909)minus119862

119903(119909) (0 le


inequalities [1]


4


6lt 4lt 2lt120587

2

(75)


2119904+1(119904 =



1198792119904+1

=(minus1)119904

1205871198612119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198792119904minus2119901+1

(76)

119904 = 1 2 for 119904 = 1 the term (1205872

6) 1198791declines


1198792119904+1

= 120577 (2119904 + 1) =1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905 minus 1119889119905

=minus1

(2119904)Ψ(2119904)

(1) (119904 = 1 2 )

(77)



1198762119904+1

=(minus1)119904

1205871198622119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198762119904minus2119901+1

(78)

119904 = 1 2 for 119904 = 1 the term (1205872

6) 1198761declines


1198762119904+1

= (2minus2119904

minus 1) 120577 (2119904 + 1) =minus1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905 + 1119889119905

(119904 = 1 2 )

(79)




2119904+1minus 1198762119904+1

= (1205872) 2119904(119904 = 1 2


2119904=2(minus1)119904

12058721198602119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)2119904minus2119901

(119904 = 1 2 )

(80)



1198602119904+1

(120587) = (minus1)119904

119904

sum

119901=1

(minus1)119901+1

1205872119901

2 (2119901 minus 1)2119904minus2119901+2

(119904 = 1 2 )

(81)









119903(119909) and 119862


119903 119860119903(1205872) 119861

119903(1205872) 119862

119903(1205872)


119903(120587) 119861

119903(120587) 119862

119903(120587)



1198781119898

=

119904

sum

119895=119898


120587119895119901

(2119895)1198702119904minus2119895+1

1198782119898

=

119904

sum

119895=119898


(2119895 minus 1)(120587

2)

2119895minus1

1198702119904minus2119895+1

(82)

respectively


100381610038161003816100381611987811198981003816100381610038161003816 = 119874(

1205872119898

(2119898))

100381610038161003816100381611987821198981003816100381610038161003816 = 119874((

120587

2)

2119898minus11

(2119898 minus 1))

(83)



100381610038161003816100381611987811198981003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119904

sum

119895=119898


120587119895119901

(2119895)1198702119904minus2119895+1

10038161003816100381610038161003816100381610038161003816100381610038161003816

le120587

2

119904

sum

119895=119898

1205872119895

(2119895)=120587

2

1205872119898

(2119898)

times (1 +1205872

(2119898 + 1) (2119898 + 2)+ sdot sdot sdot +

1205872119904minus2119898

(2119898 + 1) sdot sdot sdot (2119904))

le1205872119898+1

2 (2119898)(1 +

10

56+102


le1205872119898+1

2 (2119898)(1 +

1

3+1


=1205872119898+1

2 (2119898)

3

2le31205872119898

(2119898)

(84)


m = 12 Array [K m] K1=120587

2 d [n

minus xminus] =

xn

n

For [s = 1 s le m2 s + +

K2s+1 =

1

2((minus1)

sd [2s + 1 120587] +s

sum

p=1

K2s+1minus2p


(2p))

K2s = (minus1)

sd [2s120587

2] +

s

sum

p=1

K2s+1minus2p


2)

2pminus1

(2p minus 1)

]



Consequently |1198781119898| = 119874(120587

2119898


100381610038161003816100381611987821198981003816100381610038161003816 le 2(

120587

2)

2119898minus11

(2119898 minus 1)

so 100381610038161003816100381611987821198981003816100381610038161003816 = 119874((

120587

2)

2119898minus11

(2119898 minus 1))

(85)


119903with a




119903for



infin

]=0((minus1)](119903+1)

(2] + 1)119903+1




119903(119909) 119861

119903(119909) and 119862

119903(119909) for any 119909 Their




Acknowledgments


References












119903






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119903 119861119903

and 119862119903in (4) for a given 119909




1198602119904minus1

(119909) = 2(minus1)119904

infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904

+120587

2

119904minus1

sum

119901=1

(minus1)119901

21199011198632119904minus2119901minus1

(119909)

1198602119904(119909) = minus 2(minus1)

119904

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904+1

+120587

2

119904

sum

119901=1

(minus1)119901

21199011198632119904minus2119901

(119909)

(65)

119904 = 1 2 3 (0 le 119909 le 120587) 1198630= 1 for 119904 = 0119860

0(119909) =


infin

sum

V=0

sin [(2V + 1) 119909](2V + 1)2119904

=1

2

infin

sum

119899=1

sin 1198991199091198992119904

minus

infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905

(1198902119905

+ 1) sin119909

(1198902119905 + 1)2

minus 41198902119905cos2119909119889119905

infin

sum

V=0

cos [(2V + 1) 119909](2V + 1)2119904+1

=1

2

infin

sum

119899=1

cos 1198991199091198992119904+1

minus

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904+1

=1

Γ (2119904+1)int

infin

0

1199052119904

119890119905

(1198902119905

minus 1) cos119909

(1198902119905+1)2

minus41198902119905cos2119909119889119905

(66)




1198612119904minus1

(119909)

= (minus1)119904

infin

sum

119899=1

sin 1198991199091198992119904

+

119904minus1

sum

119901=1

(minus1)119901

1198792119904minus2119901+1

1198632119901minus1

(119909)

1198612119904minus2

(119909)

= (minus1)119904

infin

sum

119899=1

cos 1198991199091198992119904minus1

+

119904minus1

sum

119901=1

(minus1)119901

1198792119904minus2119901+1

1198632119901minus2

(119909)

(67)

119904 = 1 2 3 (0 le 119909 le 2120587) 1198630= 1 for 119904 = 1119879

1119863119896(119896 = 0 1)



sum

119899=1

sin 1198991199091198992119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905 sin119909

1 minus 2119890119905 cos119909 + 1198902119905119889119905

(119909 = 0 119904 = 1 2 3 )

infin

sum

119899=1

cos 1198991199091198992119904minus1

=1


infin

0

1199052119904minus2

(119890119905 cos119909 minus 1)

1 minus 2119890119905 cos119909 + 1198902119905119889119905

(119909 = 0 119904 = 1 2 3 )

(68)



sum

119899=1

(minus1)119899cos 119899119909119899


1198761= minus ln 2

(69)



1198622119904minus1

(119909)

= (minus1)119904

infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901+1

1198632119901minus1

(119909)

1198622119904minus2

(119909)

= (minus1)119904

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904minus1

+

119904minus1

sum

119901=1

(minus1)119901

1198762119904minus2119901+1

1198632119901minus2

(119909)

(70)



infin

sum

119899=1

(minus1)119899sin 1198991199091198992119904

=minus1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905 sin119909

1 + 2119890119905 cos119909 + 1198902119905119889119905

(119904 = 1 2 3 )

infin

sum

119899=1

(minus1)119899cos 1198991199091198992119904minus1

=minus1


infin

0

1199052119904minus2

(119890119905 cos119909 + 1)

1 + 2119890119905 cos119909 + 1198902119905119889119905

(119904 = 1 2 )

(71)


sum

V=0

(minus1)V sin [(2V + 1) 119909]

2V + 1=minus1

2ln(tan(120587

4minus119909

2))

(minus120587

2lt 119909 lt

120587

2)

(72)




int

1205872minus119909

0

1198602119904minus1

(119906) 119889119906 = minus 2(minus1)119904

infin

sum

V=0

(minus1)V sin [(2V + 1) 119909](2V + 1)2119904+1

+120587

2

119904

sum

119901=1

(minus1)119901

21199011198632119904minus2119901

(120587

2minus 119909)

int

1205872minus119909

0

1198602119904minus2

(119906) 119889119906 = 2(minus1)119904

infin

sum

V=0

(minus1)V cos [(2V + 1) 119909]

(2V + 1)2119904

minus120587

2

119904minus1

sum

119901=1

(minus1)119901

21199011198632119904minus2119901minus1

(120587

2minus 119909)

(73)

119904 = 1 2 3 (minus1205872 le 119909 le 1205872) 1198630(119909) = 1 for 119904 = 1


infin

sum

V=0

(minus1)V sin [(2V + 1) 119909](2V + 1)2119904+1

=1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905

(1198902119905

minus 1) sin119909

(1198902119905 + 1)2

minus 41198902119905sin2119909119889119905

(119904 = 1 2 )

infin

sum

V=0

(minus1)V cos [(2V + 1) 119909]

(2V + 1)2119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905

(1198902119905

+ 1) cos119909

(1198902119905 + 1)2

minus 41198902119905sin2119909119889119905

(119904 = 1 2 )

(74)



119903(119909)minus119862

119903(119909) (0 le


inequalities [1]


4


6lt 4lt 2lt120587

2

(75)


2119904+1(119904 =



1198792119904+1

=(minus1)119904

1205871198612119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198792119904minus2119901+1

(76)

119904 = 1 2 for 119904 = 1 the term (1205872

6) 1198791declines


1198792119904+1

= 120577 (2119904 + 1) =1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905 minus 1119889119905

=minus1

(2119904)Ψ(2119904)

(1) (119904 = 1 2 )

(77)



1198762119904+1

=(minus1)119904

1205871198622119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198762119904minus2119901+1

(78)

119904 = 1 2 for 119904 = 1 the term (1205872

6) 1198761declines


1198762119904+1

= (2minus2119904

minus 1) 120577 (2119904 + 1) =minus1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905 + 1119889119905

(119904 = 1 2 )

(79)




2119904+1minus 1198762119904+1

= (1205872) 2119904(119904 = 1 2


2119904=2(minus1)119904

12058721198602119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)2119904minus2119901

(119904 = 1 2 )

(80)



1198602119904+1

(120587) = (minus1)119904

119904

sum

119901=1

(minus1)119901+1

1205872119901

2 (2119901 minus 1)2119904minus2119901+2

(119904 = 1 2 )

(81)









119903(119909) and 119862


119903 119860119903(1205872) 119861

119903(1205872) 119862

119903(1205872)


119903(120587) 119861

119903(120587) 119862

119903(120587)



1198781119898

=

119904

sum

119895=119898


120587119895119901

(2119895)1198702119904minus2119895+1

1198782119898

=

119904

sum

119895=119898


(2119895 minus 1)(120587

2)

2119895minus1

1198702119904minus2119895+1

(82)

respectively


100381610038161003816100381611987811198981003816100381610038161003816 = 119874(

1205872119898

(2119898))

100381610038161003816100381611987821198981003816100381610038161003816 = 119874((

120587

2)

2119898minus11

(2119898 minus 1))

(83)



100381610038161003816100381611987811198981003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119904

sum

119895=119898


120587119895119901

(2119895)1198702119904minus2119895+1

10038161003816100381610038161003816100381610038161003816100381610038161003816

le120587

2

119904

sum

119895=119898

1205872119895

(2119895)=120587

2

1205872119898

(2119898)

times (1 +1205872

(2119898 + 1) (2119898 + 2)+ sdot sdot sdot +

1205872119904minus2119898

(2119898 + 1) sdot sdot sdot (2119904))

le1205872119898+1

2 (2119898)(1 +

10

56+102


le1205872119898+1

2 (2119898)(1 +

1

3+1


=1205872119898+1

2 (2119898)

3

2le31205872119898

(2119898)

(84)


m = 12 Array [K m] K1=120587

2 d [n

minus xminus] =

xn

n

For [s = 1 s le m2 s + +

K2s+1 =

1

2((minus1)

sd [2s + 1 120587] +s

sum

p=1

K2s+1minus2p


(2p))

K2s = (minus1)

sd [2s120587

2] +

s

sum

p=1

K2s+1minus2p


2)

2pminus1

(2p minus 1)

]



Consequently |1198781119898| = 119874(120587

2119898


100381610038161003816100381611987821198981003816100381610038161003816 le 2(

120587

2)

2119898minus11

(2119898 minus 1)

so 100381610038161003816100381611987821198981003816100381610038161003816 = 119874((

120587

2)

2119898minus11

(2119898 minus 1))

(85)


119903with a




119903for



infin

]=0((minus1)](119903+1)

(2] + 1)119903+1




119903(119909) 119861

119903(119909) and 119862

119903(119909) for any 119909 Their




Acknowledgments


References












119903






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











int

1205872minus119909

0

1198602119904minus1

(119906) 119889119906 = minus 2(minus1)119904

infin

sum

V=0

(minus1)V sin [(2V + 1) 119909](2V + 1)2119904+1

+120587

2

119904

sum

119901=1

(minus1)119901

21199011198632119904minus2119901

(120587

2minus 119909)

int

1205872minus119909

0

1198602119904minus2

(119906) 119889119906 = 2(minus1)119904

infin

sum

V=0

(minus1)V cos [(2V + 1) 119909]

(2V + 1)2119904

minus120587

2

119904minus1

sum

119901=1

(minus1)119901

21199011198632119904minus2119901minus1

(120587

2minus 119909)

(73)

119904 = 1 2 3 (minus1205872 le 119909 le 1205872) 1198630(119909) = 1 for 119904 = 1


infin

sum

V=0

(minus1)V sin [(2V + 1) 119909](2V + 1)2119904+1

=1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905

(1198902119905

minus 1) sin119909

(1198902119905 + 1)2

minus 41198902119905sin2119909119889119905

(119904 = 1 2 )

infin

sum

V=0

(minus1)V cos [(2V + 1) 119909]

(2V + 1)2119904

=1

Γ (2119904)int

infin

0

1199052119904minus1

119890119905

(1198902119905

+ 1) cos119909

(1198902119905 + 1)2

minus 41198902119905sin2119909119889119905

(119904 = 1 2 )

(74)



119903(119909)minus119862

119903(119909) (0 le


inequalities [1]


4


6lt 4lt 2lt120587

2

(75)


2119904+1(119904 =



1198792119904+1

=(minus1)119904

1205871198612119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198792119904minus2119901+1

(76)

119904 = 1 2 for 119904 = 1 the term (1205872

6) 1198791declines


1198792119904+1

= 120577 (2119904 + 1) =1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905 minus 1119889119905

=minus1

(2119904)Ψ(2119904)

(1) (119904 = 1 2 )

(77)



1198762119904+1

=(minus1)119904

1205871198622119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)1198762119904minus2119901+1

(78)

119904 = 1 2 for 119904 = 1 the term (1205872

6) 1198761declines


1198762119904+1

= (2minus2119904

minus 1) 120577 (2119904 + 1) =minus1

Γ (2119904 + 1)int

infin

0

1199052119904

119890119905 + 1119889119905

(119904 = 1 2 )

(79)




2119904+1minus 1198762119904+1

= (1205872) 2119904(119904 = 1 2


2119904=2(minus1)119904

12058721198602119904+1

(120587) +

119904minus1

sum

119901=1

(minus1)119901+1

1205872119901

(2119901 + 1)2119904minus2119901

(119904 = 1 2 )

(80)



1198602119904+1

(120587) = (minus1)119904

119904

sum

119901=1

(minus1)119901+1

1205872119901

2 (2119901 minus 1)2119904minus2119901+2

(119904 = 1 2 )

(81)









119903(119909) and 119862


119903 119860119903(1205872) 119861

119903(1205872) 119862

119903(1205872)


119903(120587) 119861

119903(120587) 119862

119903(120587)



1198781119898

=

119904

sum

119895=119898


120587119895119901

(2119895)1198702119904minus2119895+1

1198782119898

=

119904

sum

119895=119898


(2119895 minus 1)(120587

2)

2119895minus1

1198702119904minus2119895+1

(82)

respectively


100381610038161003816100381611987811198981003816100381610038161003816 = 119874(

1205872119898

(2119898))

100381610038161003816100381611987821198981003816100381610038161003816 = 119874((

120587

2)

2119898minus11

(2119898 minus 1))

(83)



100381610038161003816100381611987811198981003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119904

sum

119895=119898


120587119895119901

(2119895)1198702119904minus2119895+1

10038161003816100381610038161003816100381610038161003816100381610038161003816

le120587

2

119904

sum

119895=119898

1205872119895

(2119895)=120587

2

1205872119898

(2119898)

times (1 +1205872

(2119898 + 1) (2119898 + 2)+ sdot sdot sdot +

1205872119904minus2119898

(2119898 + 1) sdot sdot sdot (2119904))

le1205872119898+1

2 (2119898)(1 +

10

56+102


le1205872119898+1

2 (2119898)(1 +

1

3+1


=1205872119898+1

2 (2119898)

3

2le31205872119898

(2119898)

(84)


m = 12 Array [K m] K1=120587

2 d [n

minus xminus] =

xn

n

For [s = 1 s le m2 s + +

K2s+1 =

1

2((minus1)

sd [2s + 1 120587] +s

sum

p=1

K2s+1minus2p


(2p))

K2s = (minus1)

sd [2s120587

2] +

s

sum

p=1

K2s+1minus2p


2)

2pminus1

(2p minus 1)

]



Consequently |1198781119898| = 119874(120587

2119898


100381610038161003816100381611987821198981003816100381610038161003816 le 2(

120587

2)

2119898minus11

(2119898 minus 1)

so 100381610038161003816100381611987821198981003816100381610038161003816 = 119874((

120587

2)

2119898minus11

(2119898 minus 1))

(85)


119903with a




119903for



infin

]=0((minus1)](119903+1)

(2] + 1)119903+1




119903(119909) 119861

119903(119909) and 119862

119903(119909) for any 119909 Their




Acknowledgments


References












119903






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119903(119909) and 119862


119903 119860119903(1205872) 119861

119903(1205872) 119862

119903(1205872)


119903(120587) 119861

119903(120587) 119862

119903(120587)



1198781119898

=

119904

sum

119895=119898


120587119895119901

(2119895)1198702119904minus2119895+1

1198782119898

=

119904

sum

119895=119898


(2119895 minus 1)(120587

2)

2119895minus1

1198702119904minus2119895+1

(82)

respectively


100381610038161003816100381611987811198981003816100381610038161003816 = 119874(

1205872119898

(2119898))

100381610038161003816100381611987821198981003816100381610038161003816 = 119874((

120587

2)

2119898minus11

(2119898 minus 1))

(83)



100381610038161003816100381611987811198981003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119904

sum

119895=119898


120587119895119901

(2119895)1198702119904minus2119895+1

10038161003816100381610038161003816100381610038161003816100381610038161003816

le120587

2

119904

sum

119895=119898

1205872119895

(2119895)=120587

2

1205872119898

(2119898)

times (1 +1205872

(2119898 + 1) (2119898 + 2)+ sdot sdot sdot +

1205872119904minus2119898

(2119898 + 1) sdot sdot sdot (2119904))

le1205872119898+1

2 (2119898)(1 +

10

56+102


le1205872119898+1

2 (2119898)(1 +

1

3+1


=1205872119898+1

2 (2119898)

3

2le31205872119898

(2119898)

(84)


m = 12 Array [K m] K1=120587

2 d [n

minus xminus] =

xn

n

For [s = 1 s le m2 s + +

K2s+1 =

1

2((minus1)

sd [2s + 1 120587] +s

sum

p=1

K2s+1minus2p


(2p))

K2s = (minus1)

sd [2s120587

2] +

s

sum

p=1

K2s+1minus2p


2)

2pminus1

(2p minus 1)

]



Consequently |1198781119898| = 119874(120587

2119898


100381610038161003816100381611987821198981003816100381610038161003816 le 2(

120587

2)

2119898minus11

(2119898 minus 1)

so 100381610038161003816100381611987821198981003816100381610038161003816 = 119874((

120587

2)

2119898minus11

(2119898 minus 1))

(85)


119903with a




119903for



infin

]=0((minus1)](119903+1)

(2] + 1)119903+1




119903(119909) 119861

119903(119909) and 119862

119903(119909) for any 119909 Their




Acknowledgments


References












119903






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










m = 12 Array [K m] K1=120587

2 d [n

minus xminus] =

xn

n

For [s = 1 s le m2 s + +

K2s+1 =

1

2((minus1)

sd [2s + 1 120587] +s

sum

p=1

K2s+1minus2p


(2p))

K2s = (minus1)

sd [2s120587

2] +

s

sum

p=1

K2s+1minus2p


2)

2pminus1

(2p minus 1)

]



Consequently |1198781119898| = 119874(120587

2119898


100381610038161003816100381611987821198981003816100381610038161003816 le 2(

120587

2)

2119898minus11

(2119898 minus 1)

so 100381610038161003816100381611987821198981003816100381610038161003816 = 119874((

120587

2)

2119898minus11

(2119898 minus 1))

(85)


119903with a




119903for



infin

]=0((minus1)](119903+1)

(2] + 1)119903+1




119903(119909) 119861

119903(119909) and 119862

119903(119909) for any 119909 Their




Acknowledgments


References












119903






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















119903






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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