Fractional Hypergeometric Functions - MDPI

��

Citation: Jain, S.; Cattani, C.;

Agarwal, P. Fractional

Hypergeometric Functions. Symmetry

2022, 14, 714. https://doi.org/

10.3390/sym14040714

Academic Editors: Sergei D.

Odintsov and Alexander Zaslavski

Received: 22 December 2021

Accepted: 7 March 2022

Published: 1 April 2022

Publisher’s Note: MDPI stays neutral

with regard to jurisdictional claims in

published maps and institutional affil-

iations.

Copyright: © 2022 by the authors.

Licensee MDPI, Basel, Switzerland.

This article is an open access article

distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

symmetryS S

Review

Fractional Hypergeometric FunctionsShilpi Jain 1,* , Carlo Cattani 2 and Praveen Agarwal 3,4

1 Department of Mathematics, Poornima College of Engineering, Jaipur 302022, India2 Engineering School (DEIM), University of Tuscia, 01100 Viterbo, Italy; [email protected] Applied Non-Liner Science Lab, Department of Mathematics, Anand International College of Engineering,

Jaipur 303012, India4 Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman 20550, United Arab Emirates;

[email protected]* Correspondence: [email protected]

Abstract: The fractional calculus of special functions has significant importance and applications invarious fields of science and engineering. Here, we aim to find the fractional integral and differen-tial formulas of the extended hypergeometric-type functions by using the Marichev–Saigo–Maedaoperators. All the outcomes presented here are of general attractiveness and can yield a number ofprevious works as special cases due to the high degree of symmetry of the involved functions.

Keywords: fractional integral and differential operators; extended hypergeometric function; pochham-mer symbol; gamma function

1. Introduction

Fractional calculus is nowadays one of the most rapidly growing subjects of mathemat-ical analysis, in spite of the fact that it is nearly 300 years old. The giants of mathematics,G.W. Leibnitz and L. Euler, thought about the possibility to perform differentiation ofnon-integer order. The real birth and far-reaching development of fractional calculus aredue to the numerous efforts of mathematicians from the XIX century to the beginning of theXX century. A current trend in contemporary fractional calculus operators is the generaliza-tion of fractional calculus operators (GFCO). Due to the expansion of numerous and evenunexpected recent applications of the operators of the classical fractional calculus operators,GFCO has become a very powerful tool stimulating the development of this field.

Since then, many known researchers and applied scientists have made valuable contri-butions in the modifications of the fractional calculus operators and their generalizations.The generalized Marichev–Saigo–Maeda fractional integral was introduced by Marichev [1]as Mellin-type convolution operators with the Appell function F3 in their kernel. Theseoperators were rediscovered and studied by Saigo [2] and, subsequently, by Saigo andMaeda [3] as generalizations of the Saigo fractional integral operators, which were firststudied by Saigo [4] and then applied by Srivastava and Saigo [5], Srivastava and Agar-wal [6] and Choi and Agarwal [7] in their systematic investigations. On the other hand,Virchenko et al. [8] defined some fractional integral operators involving the generalizedfunction r

˜̃F(z) and their inverses in terms of the Mellin transformation. At the same time,some researchers studied the fractional operators associated with Mittag-Leffler-type func-tions [9,10] and some studied extended Caputo-type and Erdlyi–Kober-type fractionalderivative operators [11–13]. We can find more details in this direction in the survey papersby Kiryakova [14,15] and associated books [16–18].

2. Preliminaries

In recent years, many extensions and generalizations of special functions have wit-nessed a significant evolution. This modification in the theory of special functions offers an

Symmetry 2022, 14, 714. https://doi.org/10.3390/sym14040714 https://www.mdpi.com/journal/symmetry

https://doi.org/10.3390/sym14040714


https://creativecommons.org/

https://creativecommons.org/licenses/by/4.0/

https://creativecommons.org/licenses/by/4.0/

https://www.mdpi.com/journal/symmetry

https://www.mdpi.com

https://orcid.org/0000-0002-0906-2801

https://orcid.org/0000-0002-7504-0424

https://orcid.org/0000-0001-7556-8942


https://www.mdpi.com/journal/symmetry

https://www.mdpi.com/article/10.3390/sym14040714?type=check_update&version=1

Symmetry 2022, 14, 714 2 of 15

analytic foundation base for the many problems in mathematical physics and engineeringsciences, which have been solved absolutely and which have various practical applications.The theory of special functions revolves around the two most important basic special func-tions, i.e., the beta function and the Gamma function, because most of the special functionsare expressed either in terms of these functions.

Classical Euler beta function defined as follows [19].

B(x1, x2) =∫ 1

0tx1−1(1− t)x2−1dt, <(x1),<(x2) > 0. (1)

Gamma function defined as follows [19].

Γ(x1) =∫ ∞

0tx1−1e−tdt, <(x1) > 0. (2)

Furthermore, the mathematical and physical applications of hypergeometric functions canbe found in various areas of applied mathematics, mathematical physics, and engineering.The Gauss hypergeometric function is a solution of a homogenous second-order differentialequation, which is called the hypergeometric differential equation, and it is given by

z(1− z)d2wdz2 + (c− (a + b + 1)z)

dwdz− abw = 0 (3)

The Gauss hypergeometric function 2F1 is defined as [20]

2F1(a, b, c; z) = F(a, b, c; z) =∞

∑k=0

(a)k(b)k(c)k

zk

k!, (4)

where (u)k represents the Pochhammer symbol defined below:

(u)k :=Γ(u + k)

Γ(u)=

{1 k = 0; u ∈ C \ {0},u(u + 1) · · · (u + k− 1) k ∈ N; u ∈ C.

Later, Kummer replaced the parameter z by zb and took the limit b → ∞ in the (3); then,

the hypergeometric differential equation becomes a confluent hypergeometric differentialequation or Kummer’s equation.

zd2wdz2 + (c− z)

dwdz− aw = 0 (5)

The confluent hypergeometric function is the solution of the above differential Equation (5).The confluent hypergeometric function is defined as [20]

1F1(a, c; z) = Φ(a, c; z) =∞

∑k=0

(a)k(c)k

zk

k!. (6)

Very recently, Goyal et al. [21] introduced an extension of the beta function using theWiman function, thus studying the various properties and relationships of that function:

B(u)(u1,u2)

(y1, y2) =∫ 1

0ty1−1(1− t)y2−1Eu1,u2

(−u(t(1− t))−1

)dt, (7)

where min{<(y1),<(y2)} > 0, <(u1) > 0,<(u2) > 0, u ≥ 0 and Eu1,u2(z) is 2-parameterMittag-Leffler function given by [22].

Motivated by the above work, Jain et al. [23] extended the Gauss hypergeometricfunction and confluent hypergeometric function by using the above extended beta functionand studied the various properties of these extended functions. They also studied the

Symmetry 2022, 14, 714 3 of 15

increasing or decreasing nature (monotonicity), log-concavity, and log-convexity of theextended beta function defined in [21].

The extended Gauss hypergeometric function is defined as [23]:

F(s)(s1,s2)

(q0, q1, q2; z) =∞

∑k=0

B(s)(s1,s2)

(q1 + k, q2 − p1)

B(q1, q2 − q1)(q0)k

zk

k!, (8)

where <(q2) > <(q1) > 0, <(s1) > 0,<(s2) > 0, s ≥ 0, |z| < 1 and B(s)(s1,s2)

(w1, w2) is theextended beta function.

The extended confluent hypergeometric function is defined as [23]:

Φ(s)(s1,s2)

(q1, q2; z) =∞

∑k=0

B(s)(s1,s2)

(q1 + k, q2 − q1)

B(q1, q2 − q1)

zk

k!, (9)

where (<(q2) > <(q1) > 0, <(s1) > 0,<(s2) > 0 and s ≥ 0).

Remark 1. If we set s1 = s2 = 1 and s = 0 , then the extended Gauss hypergeometric functionF(s)(s1,s2)

(q0, q1, q2; z) (8), and the extended confluent hypergeometric function Φ(s)(s1,s2)

(q1, q2; z) (9),reduced to the Gauss hypergeometric function 2F1(q0, q1, q2; z) (4) and the confluent hypergeometricfunction Φ(q1, q2; z) (6).

The concept of the Hadamard product (convolution) of the functions f and g is veryimportant for our results.

The Hadamard product of f and g can be defined as follows [24]:

( f ∗ g)(z) =∞

∑n=0

anbnzn = (g ∗ f )(z), (10)

where f (z) = ∑∞n=0 anzn and g(z) = ∑∞

n=0 bnzn.The third Appell function F3 (also known as one of the functions in Horn’s list) is

defined as follows (see [25]):

F3(a, a′, b, b′; c; x; y) =∞

∑m,n=0

(a)m(a)n(b)m(b′)n

(c)m+n

xm

m!yn

n! (11)

here, (max | x |, | y | < 1)Now, we recall here the generalized Marichev–Saigo–Maeda fractional integral and

fractional derivative operators, introduced by Marichev [1], as Mellin-type convolutionoperators with an Appell function F3 in their kernel, which are defined as follows:

Definition 1. Let a, a′, b, b′, c ∈ C and x > 0, then for <(c) > 0(Ia,a′ ,b,b′ ,c0,x f

)(x)

=x−a

Γ(c)

∫ x

0(x− t)c−1t−a′F3

(a, a′, b, b′; c; 1− t

x, 1− x

t

)f (t)dt,

(12)

and(Ia,a′ ,b,b′ ,cx,∞ f

)(x)

=x−a′

Γ(c)

∫ ∞

x(t− x)c−1t−aF3

(a, a′, b, b′; c; 1− x

t, 1− t

x

)f (t)dt

(13)

provided the function f (t) is constrained such that the integrals in (12) and (13) exist.

Symmetry 2022, 14, 714 4 of 15

The above fractional integral operators in Equations (12) and (13) can be written as

(Ia,a′ ,b,b′ ,c0,x f

)(x) =

(d

dx

)k(Ia,a′ ,b+k,b′ ,c+k0,x f

)(x)

(<(c) ≤ 0; k = [−<(c) + 1])(14)

and (Ia,a′ ,b,b′ ,cx,∞ f

)(x) =

(− d

dx

)k(Ia,a′ ,b,b′+k,c+kx,∞ f

)(x)

(<(c) ≤ 0; k = [−<(c) + 1])(15)

Furthermore, the corresponding Marichev–Saigo–Maeda fractional differential operatorsare given as:

Definition 2. Let a, a′, b, b′, c ∈ C and x > 0, then(Da,a′ ,b,b′ ,c

0,x f)(x) =

(I−a′ ,−a,−b′ ,−b,−c0,x f

)(x)

=

(d

dx

)k(I−a′ ,−a,−b′+k,−b,−c+k0,x f

)(x) (<(c) > 0; k = [<(c)] + 1)

=1

Γ(k− c)

(d

dx

)k(x)a′

∫ x

0(x− t)k−c−1ta

×F3

(−a′,−a, k− b′,−b; k− c; 1− t

x, 1− x

t

)f (t)dt

(16)

and(Da,a′ ,b,b′ ,c

x,∞ f)(x) =

(I−a′ ,−a,−b′ ,−b,−cx,∞ f

)(x)

=

(− d

dx

)k(I−a′ ,−a,−b′ ,−b+k,−c+kx,∞ f

)(x) (<(c) > 0; k = [<(c)] + 1)

=1

Γ(k− c)

(− d

dx

)k(x)a

∫ ∞

x(t− x)k−c−1ta′

×F3

(−a′,−a,−b′, k− b; k− c; 1− x

t, 1− t

x

)f (t)dt

(17)

Remark 2. Here, it is important to mention that the Appell function F3 involved in Definitions (1) and (2)satisfies a system of two linear partial differential equations of the second order and reduces them tothe Gauss hypergeometric function 2F1 as follows (see ([26], p. 25, Equation (35)) and ([27], p. 301,Equation (9.4)):

F3(a, c− a, b, c− b; c; x; y) =2 F1

[a, bc

; x + y− xy]

(18)

It is easily observed that

F3(a, 0, b, b′; c; x; y) = F3(a, a′, b, 0; c; x; y) =2 F1

[a, bc

; x]

(19)

and

F3(0, a′, b, b′; c; x; y) = F3(a, a′, 0, b′; c; x; y) =2 F1

[a′, b′

c; y]

(20)

Symmetry 2022, 14, 714 5 of 15

In view of the obvious reduction formulas with first replacing the parameter a by a + b , and thensetting a′ = b′ = 0, b = −d and c = a in Equations (12), (13), (16) and (17) the generalizedMarichev–Saigo–Maeda fractional integral and fractional derivative operators can be reduced tothe Saigo fractional integral and derivative operators Ia,b,d

0,x , Ia,b,dx,∞ , Da,b,d

0,x , and Da,b,dx,∞ involving the

hypergeometric function 2F1 in their kernel, which is defined by

(Ia,b,d0,x f

)(x) =

x−a−b

Γ(a)

∫ x

0(x− t)a−1

2 F1

(a + b,−d; a; 1− t

x

)f (t)dt, (21)

and (Ia,b,dx,∞ f

)(x) =

1Γ(a)

∫ ∞

x(t− x)a−1

2 F1

(a + b,−d; a; 1− x

t

)f (t)dt (22)

(Da,b,d

0,x f)(x) =

(I−a,−b,a+d0,x f

)(x) (23)

and (Da,b,d

x,∞ f)(x) =

(I−a,−b,a+dx,∞ f

)(x), (24)

where a, b, d ∈ C and x > 0.Several further consequences of the Marichev–Saigo–Maeda fractional integral and fractional

derivative operators can easily be derived by first setting b = −a and b = 0 in Equations (21) and (22).We easily obtain the interesting consequences of the Marichev–Saigo–Maeda fractional integral andfractional derivative operators would involve the Erdélyi–Kober fractional integral operators E a,d

0,x

and Ka,dx,∞, the Riemann–Liouville fractional integral operatorRa

0,x and the Weyl fractional integraloperatorW a

x,∞, respectively:

(Ra

0,x f)(x) =

1Γ(a)

∫ x

0(x− t)a−1 f (t)dt =

(Ia,−a,d0,x f

)(x), (25)

and (W a

x,∞ f)(x) =

1Γ(a)

∫ ∞

x(t− x)a−1 f (t)dt =

(Ia,−a,dx,∞ f

)(x) (26)

and (E a,d

0,x f)(x) =

x−a−d

Γ(a)

∫ x

0(x− t)a−1 td f (t)dt =

(Ia,0,d0,x f

)(x), (27)

and (Ka,d

x,∞ f)(x) =

xd

Γ(a)

∫ ∞

x(t− x)a−1 t−a−d f (t)dt =

(Ia,0,dx,∞ f

)(x) (28)

The following lemmas proved in [4] are useful to prove our main results.

Lemma 1 ([4]). Let a, a′, b, b, c and d ∈ C, x > 0. Then, the Marichev–Saigo–Maeda fractionalintegral of the power td−1 is given as follows:

1. If <(c) > 0 and <(d) > max{0,<(a + a′ + b− c),<(a′ − b′)}, then(Ia,a′ ,b,b′ ,c0,x td−1

)(x)

=Γ(d)Γ(d + c− a− a′ − b)Γ(d + b′ − a′)

Γ(d + b′)Γ(d + c− a− a′)Γ(d + c− a′ − b)xd+c−a−a′−1

(29)

Symmetry 2022, 14, 714 6 of 15

2. If <(c) > 0 and <(d) < 1 + min{<(−b),<(a + a′ − c),<(a + b′ − c)}, then(Ia,a′ ,b,b′ ,cx,∞ td−1

)(x)

=Γ(1− d− b)Γ(1− d− c + a + a′)Γ(1− d− c + a + b′)

Γ(1− d)Γ(1− d− c + a + a′ + b′)Γ(1− d + a− b)xd+c−a−a′−1

(30)

Lemma 2 ([4]). Let a, a′, b, b′, c and d ∈ C, x > 0. Then, the Marichev–Saigo–Maeda fractionalintegral of the power td−1 is given as follows:

1. If <(c) > 0 and <(d) > max{0,<(c− a− a′ − b′),<(b− a)}, then(Da,a′ ,b,b′ ,c

0,x td−1)(x)

=Γ(d)Γ(d− c + a + a′ + b′)Γ(d− b + a)

Γ(d− b)Γ(d− c + a + a′)Γ(d− c + a + b′)xd−c+a+a′−1

(31)

2. If <(c) > 0 and <(d) < 1 + min{<(b′),<(c− a− a′),<(c− a′ − b)}, then(Da,a′ ,b,b′ ,c

x,∞ td−1)(x)

=Γ(1− d + b′)Γ(1− d + c− a− a′)Γ(1− d + c− a′ − b)

Γ(1− d)Γ(1− d + c− a− a′ − b)Γ(1− d− a′ + b′)xd−c+a+a′−1

(32)

3. Main Results

3.1. Marichev–Saigo–Maeda Fractional Integral of the Function F(s)(s1,s2)

(q0, q1, q2; z) and

Φ(s)(s1,s2)

(q1, q2; z)

In this section, we establish the Marichev–Saigo–Maeda fractional integral formulasinvolving the extended Gauss hypergeometric function and extended confluent hyperge-ometric function; the results are expressed as the Hadamard product of the generalizedhypergeometric function and the extended Gauss hypergeometric function.

Theorem 1. Let x > 0, a, a′, b, b′, c, d ∈ C and <(q2) > <(q1) > 0, <(s1) > 0, <(s2) > 0,s ≥ 0, |t| < 1 be such that <(c) > 0, and <(d + n) > max{0,<(a + a′ + b− c),<(a′ − b′)};then, the following fractional integral formula holds true:(

Ia,a′ ,b,b′ ,c0,x td−1F(s)

(s1,s2)(q0, q1, q2; t)

)(x)

= xd+c−a−a′−1 Γ(d)Γ(d + c− a− a′ − b)Γ(d + b′ − a′)Γ(d + b′)Γ(d + c− a− a′)Γ(d + c− a′ − b)

×F(s)(s1,s2)

(q0, q1, q2; x) ∗ 3F3

[d, d + c− a− a′ − b, d + b′ − a′;

d + b′, d + c− a− a′, d + c− a′ − b;x] (33)

where 3F3(x) is a special case of the generalized hypergeometric function with p = q = 3 [20].

Proof. From the definition of the Marichev–Saigo–Maeda fractional integral operator,we have:(

Ia,a′ ,b,b′ ,c0,x f (t)

)(x) =

x−a

Γ(c)

∫ x

0(x− t)c−1t−a′F3

(a, a′, b, b′; c; 1− t

x, 1− x

t

)f (t)dt, (34)

Then, taking f (t) = td−1F(s)(s1,s2)

(q0, q1, q2; t) in the above equation, and using the definitionof extended Gauss hypergeometric function (8), we obtain:

(Ia,a′ ,b,b′ ,c0,x td−1F(s)

(s1,s2)(q0, q1, q2; t)

)(x) =

(Ia,a′ ,b,b′ ,c0,x td−1

( ∞

∑n=0

B(s)(s1,s2)

(q1 + n, q2 − p1)

B(q1, q2 − q1)(q0)n

zn

n!

)). (35)

Symmetry 2022, 14, 714 7 of 15

Then, interchanging the order of integration and summation, we have:


(s1,s2)(q0, q1, q2; t)

)(x) =

∞

∑n=0

(q0)nB(s)(s1,s2)

(q1 + n, q2 − p1)

B(q1, q2 − q1)

1n!

(Ia,a′ ,b,b′ ,c0,x td+n−1

)(x). (36)

Then, by the using Lemma (1), with d replaced by d + n, we have:


(s1,s2)(q0, q1, q2; t)

)(x) =

∞

∑n=0

(q0)nB(s)(s1,s2)

(q1 + n, q2 − p1)

B(q1, q2 − q1)

× Γ(d + n)Γ(d + c− a− a′ − b + n)Γ(d + b′ − a′ + n)Γ(d + b′ + n)Γ(d + c− a− a′ + n)Γ(d + c− a′ − b + n)

xd+n+c−a−a′−1

n!.

(37)

After simplification, the above equation reduces to


(s1,s2)(q0, q1, q2; t)

)(x) = xd+c−a−a′−1

∞

∑n=0

(q0)nB(s)(s1,s2)

(q1 + k, q2 − p1)

B(q1, q2 − q1)

× Γ(d + n)Γ(d + c− a− a′ − b + n)Γ(d + b′ − a′ + n)Γ(d + b′ + n)Γ(d + c− a− a′ + n)Γ(d + c− a′ − b + n)

xn

n!.

(38)

After using the (a)n = Γ(a+n)Γ(a) , the above equation reduces to the following form:(


(s1,s2)(q0, q1, q2; t)

)(x) = xd+c−a−a′−1 Γ(d)Γ(d + c− a− a′ − b)Γ(d + b′ − a′)

Γ(d + b′)Γ(d + c− a− a′)Γ(d + c− a′ − b)

×∞

∑n=0

(q0)nB(s)(s1,s2)

(q1 + n, q2 − p1)

B(q1, q2 − q1)

(d)n(d + c− a− a′ − b)n(d + b′ − a′)n

(d + b′)n(d + c− a− a′)n(d + c− a′ − b)n

xn

n!.

(39)

then by interpreting the above equation with the help of the concept of the Hadamardproduct given by (10), we obtain our desired result.(


(s1,s2)(q0, q1, q2; t)

)(x)


×F(s)(s1,s2)

(q0, q1, q2; x) ∗ 3F3

[d, d + c− a− a′ − b, d + b′ − a′;

d + b′, d + c− a− a′, d + c− a′ − b;x] (40)

Theorem 2. Let x > 0, a, a′, b, b′, c, d ∈ C and <(q2) > <(q1) > 0, <(s1) > 0, <(s2) > 0,s ≥ 0, | 1t | < 1 be such that <(c) > 0 and <(d) < 1 + min{<(−b),<(a + a′ − c),<(a + b′ −c)}; then, the following fractional integral formula holds true:(

Ia,a′ ,b,b′ ,cx,∞ td−1F(s)

(s1,s2)(q0, q1, q2;

1t)

)(x)

= xd+c−a−a′−1 Γ(1− d− b)Γ(1− d− c + a + a′)Γ(1− d− c + a + b′)Γ(1− d)Γ(1− d− c + a + a′ + b′)Γ(1− d + a− b)

×F(s)(s1,s2)

(q0, q1, q2; x) ∗ 3F3

[1− d− b, 1− d− c + a + a′, 1− d− c + a + b′;

1− d, 1− d− c + a + a′ + b′, 1− d + a− b;x] (41)


The proof of the Theorem (2) is the same as those of Theorem (1).

Symmetry 2022, 14, 714 8 of 15

Theorem 3. Let x > 0, a, a′, b, b′, c, d ∈ C and <(q2) > <(q1) > 0, <(s1) > 0, <(s2) > 0,s ≥ 0, be such that <(c) > 0, and <(d + n) > max{0,<(a + a′ + b− c),<(a′ − b′)}; then, thefollowing fractional integral formula holds true:(

Ia,a′ ,b,b′ ,c0,x td−1Φ(s)

(s1,s2)(q1, q2; t)

)(x)


×Φ(s)(s1,s2)

(q1, q2; x) ∗ 3F3

[d, d + c− a− a′ − b, d + b′ − a′;

d + b′, d + c− a− a′, d + c− a′ − b;x] (42)


Proof. From the definition of the Marichev–Saigo–Maeda fractional integral operator, wehave:

(Ia,a′ ,b,b′ ,c0,x f (t)

)(x) =

x−a

Γ(c)

∫ x

0(x− t)c−1t−a′F3

(a, a′, b, b′; c; 1− t

x, 1− x

t

)f (t)dt, (43)

Then, taking f (t) = td−1Φ(s)(s1,s2)

(q1, q2; t) in the above equation, and using the definition ofthe extended confluent hypergeometric function (9), we obtain:

(Ia,a′ ,b,b′ ,c0,x td−1Φ(s)

(s1,s2)(q1, q2; t)

)(x) =

(Ia,a′ ,b,b′ ,c0,x td−1

( ∞

∑n=0

B(s)(s1,s2)

(q1 + n, q2 − p1)

B(q1, q2 − q1)

zn

n!

)). (44)

Then, interchanging the order of integration and summation, we have:

(Ia,a′ ,b,b′ ,c0,x td−1Φ(s)

(s1,s2)(q1, q2; t)

)(x) =

∞

∑n=0

B(s)(s1,s2)

(q1 + n, q2 − p1)

B(q1, q2 − q1)

1n!

(Ia,a′ ,b,b′ ,c0,x td+n−1

)(x). (45)

Then, by similar steps tot those in the proof of Theorem (1), we obtain our desired result.(Ia,a′ ,b,b′ ,c0,x td−1Φ(s)

(s1,s2)(q1, q2; t)

)(x)


×Φ(s)(s1,s2)

(q1, q2; x) ∗ 3F3

[d, d + c− a− a′ − b, d + b′ − a′;

d + b′, d + c− a− a′, d + c− a′ − b;x] (46)

Theorem 4. Let x > 0, a, a′, b, b′, c, d ∈ C and <(q2) > <(q1) > 0, <(s1) > 0, <(s2) > 0,s ≥ 0 be such that <(c) > 0 and <(d) < 1 + min{<(−b),<(a + a′ − c),<(a + b′ − c)}; then,the following fractional integral formula holds true:(

Ia,a′ ,b,b′ ,cx,∞ td−1Φ(s)

(s1,s2)(q1, q2;

1t)

)(x)


×Φ(s)(s1,s2)

(q1, q2; x) ∗ 3F3

[1− d− b, 1− d− c + a + a′, 1− d− c + a + b′;

1− d, 1− d− c + a + a′ + b′, 1− d + a− b;x] (47)

where 3F3(x) is a special case of the generalized hypergeometric function with p = q = 3 anddefined in [20].

Symmetry 2022, 14, 714 9 of 15

The proof of Theorem (4) is the same as that of Theorem (3).

3.2. Marichev–Saigo–Maeda Fractional Derivative of the Function F(s)(s1,s2)

(q0, q1, q2; z) and

Φ(s)(s1,s2)

(q1, q2; z)

Here, we derive the Marichev–Saigo–Maeda fractional derivative formulas involvingthe extended Gauss hypergeometric function and extended confluent hypergeometric func-tion; the results are expressed as the Hadamard product of the generalized hypergeometricfunction and the extended Gauss hypergeometric function.

Theorem 5. Let x > 0, a, a′, b, b′, c, d ∈ C and <(q2) > <(q1) > 0, <(s1) > 0, <(s2) > 0,s ≥ 0, |t| < 1 be such that <(c) > 0 and <(d) > max{0,<(c− a− a′ − b′),<(b− a)}; then,the following fractional derivative formula holds true:(

Da,a′ ,b,b′ ,c0,x td−1F(s)

(s1,s2)(q0, q1, q2; t)

)(x)

= xd−c+a+a′−1 Γ(d)Γ(d− c + a + a′ + b′)Γ(d− b + a)Γ(d− b)Γ(d− c + a + a′)Γ(d− c + a + b′)

×F(s)(s1,s2)

(q0, q1, q2; x) ∗ 3F3

[d, d− c + a + a′ + b′, d− b + a;

d− b, d− c + a + a′, d− c + a + b′;x] (48)

where 3F3(.) is a special case of the generalized hypergeometric function [20].

Proof. Applying the Marichev–Saigo–Maeda fractional derivative operator defined in (16)to function f (t) = td−1F(s)

(s1,s2)(q0, q1, q2; t), and using the definition of extended Gauss

hypergeometric function (8), we obtain:

(Da,a′ ,b,b′ ,c

0,x td−1F(s)(s1,s2)

(q0, q1, q2; t))(x) =

(Da,a′ ,b,b′ ,c

0,x td−1( ∞

∑n=0

B(s)(s1,s2)

(q1 + n, q2 − p1)

B(q1, q2 − q1)(q0)n

zn

n!

)). (49)

Then, interchanging the order of integration and summation because the Marichev–Saigo–Maeda fractional differential operator exists, involving the extended hypergeometric func-tion with conditions x > 0, a, a′, b, b′, c, d ∈ C and<(q2) > <(q1) > 0,<(s1) > 0,<(s2) > 0,s ≥ 0, |t| < 1 be such that <(c) > 0 and <(d) > max{0,<(c− a− a′ − b′),<(b− a)}, sothe convergence is uniform, we have:

(Da,a′ ,b,b′ ,c

0,x td−1F(s)(s1,s2)

(q0, q1, q2; t))(x) =

∞

∑n=0

(q0)nB(s)(s1,s2)

(q1 + n, q2 − p1)

B(q1, q2 − q1)

1n!

(Da,a′ ,b,b′ ,c

0,x td+n−1)(x). (50)

Then, by using Lemma (2), and following the same procedure as in the proof of Theorem (1),we obtain our desired result.(

Da,a′ ,b,b′ ,c0,x td−1F(s)

(s1,s2)(q0, q1, q2; t)

)(x)


×F(s)(s1,s2)

(q0, q1, q2; x) ∗ 3F3

[d, d− c + a + a′ + b′, d− b + a;

d− b, d− c + a + a′, d− c + a + b′;x] (51)

Symmetry 2022, 14, 714 10 of 15

Theorem 6. Let x > 0, a, a′, b, b′, c, d ∈ C and <(q2) > <(q1) > 0, <(s1) > 0, <(s2) > 0,s ≥ 0, | 1t | < 1 be such that<(c) > 0 and<(d) < 1+min{<(b′),<(c− a− a′),<(c− a′− b)},then the following fractional derivative formula holds true:(

Da,a′ ,b,b′ ,c0,∞ td−1F(s)

(s1,s2)(q0, q1, q2;

1t)

)(x)

= xd−c+a+a′−1 Γ(1− d + b′)Γ(1− d + c− a− a′)Γ(1− d− a′ − b + c)Γ(1− d)Γ(1− d− a− a′ − b + c)Γ(1− d− a′ + b′)

×F(s)(s1,s2)

(q0, q1, q2; x) ∗ 3F3

[1− d + b′, 1− d + c− a− a′, 1− d− a′ − b + c;

1− d, 1− d− a− a′ − b + c, 1− d− a′ + b′;x] (52)

where 3F3(.) is a special case of the generalized hypergeometric function [20].


Theorem 7. Let x > 0, a, a′, b, b′, c, d ∈ C and <(q2) > <(q1) > 0, <(s1) > 0, <(s2) > 0,s ≥ 0, be such that <(c) > 0 and <(d) > max{0,<(c − a − a′ − b′),<(b − a)}, then thefollowing fractional derivative formula holds true:(

Da,a′ ,b,b′ ,c0,x td−1Φ(s)

(s1,s2)(q1, q2; t)

)(x)


×Φ(s)(s1,s2)

(q1, q2; x) ∗ 3F3

[d, d− c + a + a′ + b′, d− b + a;

d− b, d− c + a + a′, d− c + a + b′;x] (53)

where 3F3(.) is a special case of pFq(.), when p = q = 3 [20].

Proof. Applying the Marichev–Saigo–Maeda fractional derivative operator defined in (16)to function f (t) = td−1Φ(s)

(s1,s2)(q1, q2; t), and using the definition of the extended confluent

hypergeometric function (9), we obtain:

(Da,a′ ,b,b′ ,c

0,x td−1Φ(s)(s1,s2)

(q1, q2; t))(x) =

(Da,a′ ,b,b′ ,c

0,x td−1( ∞

∑n=0

B(s)(s1,s2)

(q1 + n, q2 − p1)

B(q1, q2 − q1)

zn

n!

)). (54)

Then, interchanging the order of integration and summation, which is valid under theconditions of Theorem (7), we have:

(Da,a′ ,b,b′ ,c

0,x td−1Φ(s)(s1,s2)

(q1, q2; t))(x) =

∞

∑n=0

B(s)(s1,s2)

(q1 + n, q2 − p1)

B(q1, q2 − q1)

1n!

(Da,a′ ,b,b′ ,c

0,x td+n−1)(x). (55)

Then, by using Lemma (2), and following the same procedure as in the proof of Theorem (1),we obtain our desired result.(

Da,a′ ,b,b′ ,c0,x td−1Φ(s)

(s1,s2)(q1, q2; t)

)(x)


×Φ(s)(s1,s2)

(q1, q2; x) ∗ 3F3

[d, d− c + a + a′ + b′, d− b + a;

d− b, d− c + a + a′, d− c + a + b′;x] (56)

Symmetry 2022, 14, 714 11 of 15

Theorem 8. Let x > 0, a, a′, b, b′, c, d ∈ C and <(q2) > <(q1) > 0, <(s1) > 0, <(s2) > 0,s ≥ 0, be such that <(c) > 0 and <(d) < 1 + min{<(b′),<(c− a− a′),<(c− a′ − b)} thenthe following fractional derivative formula holds true:(

Da,a′ ,b,b′ ,c0,∞ td−1Φ(s)

(s1,s2)(q1, q2;

1t)

)(x)


×Φ(s)(s1,s2)

(q1, q2; x) ∗ 3F3

[1− d + b′, 1− d + c− a− a′, 1− d− a′ − b + c;

1− d, 1− d− a− a′ − b + c, 1− d− a′ + b′;x] (57)

where 3F3(.) is a special case of pFq(.), when p = q = 3 and defined in [20].


3.3. Examples of the Marichev–Saigo–Maeda Fractional Integral of the FunctionF(s)(s1,s2)

(q0, q1, q2; z) and Φ(s)(s1,s2)

(q1, q2; z)

Due to the high degree of symmetry, we can assume that r1 = r2 = 1 and r = 0,then a new extension of the Gauss hypergeometric function (8) and a new extension ofthe confluent hypergeometric function (9) can be reduced to the Gauss hypergeometricfunction (4) and confluent hypergeometric function (6). Then, from the formulae establishedin (33), (41), (42) and (47), we have the following results.

Corollary 1. Let x > 0, a, a′, b, b′, c, d ∈ C and <(q2) > <(q1) > 0, |t| < 1 be such that<(c) > 0, and <(d + n) > max{0,<(a + a′ + b− c),<(a′ − b′)}, then the following fractionalintegral formula holds true:(

Ia,a′ ,b,b′ ,c0,x td−1

2F1(q0, q1, q2; t))(x)


×2F1(q0, q1, q2; x) ∗ 3F3

[d, d + c− a− a′ − b, d + b′ − a′;

d + b′, d + c− a− a′, d + c− a′ − b;x] (58)

where 3F3(a1, a2, a3, b1, b2, b3; x) is a special case of the generalized hypergeometric functionpFq(a1, ......,ap, b1, ......, bq; x), when p = q = 3 and defined in [20].

Corollary 2. Let x > 0, a, a′, b, b′, c, d ∈ C and <(q2) > <(q1) > 0, | 1t | < 1 be such that<(c) > 0 and <(d) < 1 + min{<(−b),<(a + a′ − c),<(a + b′ − c)}, then the followingfractional integral formula holds true:(

Ia,a′ ,b,b′ ,cx,∞ td−1

2F1(q0, q1, q2;1t)

)(x)


×2F1(q0, q1, q2; x) ∗ 3F3

[1− d− b, 1− d− c + a + a′, 1− d− c + a + b′;

1− d, 1− d− c + a + a′ + b′, 1− d + a− b;x] (59)


Symmetry 2022, 14, 714 12 of 15

Corollary 3. Let x > 0, a, a′, b, b′, c, d ∈ C and <(q2) > <(q1) > 0 be such that <(c) > 0,and <(d + n) > max{0,<(a + a′ + b− c),<(a′ − b′)}, then the following fractional integralformula holds true:(

Ia,a′ ,b,b′ ,c0,x td−1Φ(q1, q2; t)

)(x)


×Φ(q1, q2; x) ∗ 3F3

[d, d + c− a− a′ − b, d + b′ − a′;

d + b′, d + c− a− a′, d + c− a′ − b;x] (60)


Corollary 4. Let x > 0, a, a′, b, b′, c, d ∈ C and <(q2) > <(q1) > 0 be such that <(c) > 0 and<(d) < 1 + min{<(−b),<(a + a′ − c),<(a + b′ − c)}, then the following fractional integralformula holds true:(

Ia,a′ ,b,b′ ,cx,∞ td−1Φ(q1, q2;

1t)

)(x)


×Φ(q1, q2; x) ∗ 3F3

[1− d− b, 1− d− c + a + a′, 1− d− c + a + b′;

1− d, 1− d− c + a + a′ + b′, 1− d + a− b;x] (61)

where 3F3(a1, a2, a3, b1, b2, b3; x) is a special case of the generalized hypergeometric functionpFq(a1, ......, ap, b1, ......, bq; x), when p = q = 3 and defined in [20].

3.4. Examples of the Marichev–Saigo–Maeda Fractional Derivative of the FunctionF(s)(s1,s2)

(q0, q1, q2; z) and Φ(s)(s1,s2)

(q1, q2; z)

If we put r1 = r2 = 1 and r = 0, then a new extension of the Gauss hypergeomet-ric function (8) and a new extension of the confluent hypergeometric function (9) canbe reduced to the Gauss hypergeometric function (4) and the confluent hypergeometricfunction (6). Then, from the formulae established in (48), (52), (53) and (57), we have thefollowing results.

Corollary 5. Let x > 0, a, a′, b, b′, c, d ∈ C and <(q2) > <(q1) > 0, |t| < 1 be such that<(c) > 0 and <(d) > max{0,<(c − a − a′ − b′),<(b − a)}, then the following fractionalderivative formula holds true:(

Da,a′ ,b,b′ ,c0,x td−1

2F1(q0, q1, q2; t))(x)


×2F1(q0, q1, q2; x) ∗ 3F3

[d, d− c + a + a′ + b′, d− b + a;

d− b, d− c + a + a′, d− c + a + b′;x] (62)


Symmetry 2022, 14, 714 13 of 15

Corollary 6. Let x > 0, a, a′, b, b′, c, d ∈ C and <(q2) > <(q1) > 0, | 1t | < 1 be such that<(c) > 0 and <(d) < 1 + min{<(b′),<(c− a− a′),<(c− a′ − b)}, then the following frac-tional derivative formula holds true:(

Da,a′ ,b,b′ ,c0,∞ td−1

2F1(q0, q1, q2;1t)

)(x)


×2F1(q0, q1, q2; x) ∗ 3F3

[1− d + b′, 1− d + c− a− a′, 1− d− a′ − b + c;

1− d, 1− d− a− a′ − b + c, 1− d− a′ + b′;x] (63)


Corollary 7. Let x > 0, a, a′, b, b′, c, d ∈ C and <(q2) > <(q1) > 0 be such that <(c) > 0 and<(d) > max{0,<(c− a− a′ − b′),<(b− a)}, then the following fractional derivative formulaholds true: (

Da,a′ ,b,b′ ,c0,x td−1Φ(q1, q2; t)

)(x)


×Φ(q1, q2; x) ∗ 3F3

[d, d− c + a + a′ + b′, d− b + a;

d− b, d− c + a + a′, d− c + a + b′;x] (64)


Corollary 8. Let x > 0, a, a′, b, b′, c, d ∈ C and <(q2) > <(q1) > 0, <(s1) > 0, <(s2) > 0,s ≥ 0, be such that <(c) > 0 and <(d) < 1 + min{<(b′),<(c− a− a′),<(c− a′ − b)}, thenthe following fractional derivative formula holds true:(

Da,a′ ,b,b′ ,c0,∞ td−1Φ(q1, q2;

1t)

)(x)


×Φ(q1, q2; x) ∗ 3F3

[1− d + b′, 1− d + c− a− a′, 1− d− a′ − b + c;

1− d, 1− d− a− a′ − b + c, 1− d− a′ + b′;x] (65)


4. Concluding Remarks

Motivated by the demonstrated usages and the potential for application of the manyoperators in fractional calculus, and also the considerably large spectrum of special func-tions in mathematics, physics, and engineering, we have introduced here new resultsfor the Marichev–Saigo–Maeda fractional integral and Marichev–Saigo–Maeda fractionalderivative involving extended hypergeometric functions. Our results are expressed as theHadamard product of the extended hypergeometric functions and generalized hypergeo-metric functions. Some special cases of our main results are also derived easily due to thehigh degree of symmetry of the involved functions. The results presented in this paper arevery useful in the theory of approximation, which is used to find several inequalities of theMarichev–Saigo–Maeda fractional operators involving extended hypergeometric functions.We conclude this investigation by emphasizing the possibility of further research involvingq-extensions of hypergeometric functions.

Symmetry 2022, 14, 714 14 of 15

Funding: This research was funded by SERB grant number MTR/2017/000194.

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: Not applicable.

Acknowledgments: Shilpi Jain is very thankful to the funding agency SERB (project number:MTR/2017/000194) for providing the necessary financial support for the present study.

Conflicts of Interest: The authors declare no conflict of interest.

References1. Marichev, O.I. Volterra equation of Mellin convolution type with a Horn function in the kernel. Izvestiya Akademii Nauk BSSR.

Seriya Fiziko-Mat. Nauk 1974, 1, 128–129.2. Saigo, M. On generalized fractional calculus operators. In Proceedings of the Recent Advances in Applied Mathematics, Kuwait

City, Kuwait, 4–7 May 1996; Kuwait University Press: Kuwait City, Kuwait, 1996; pp. 441–450.3. Saigo, M.; Maeda, N. More generalization of fractional calculus. In Transform Methods and Special Functions (Varna, Bulgaria);

Bulgaria Academy of Sciences: Sofia, Bulgaria, 1998; pp. 386–400.4. Saigo, M. A remark on integral operators involving the Gauss hypergeometric functions. Math. Rep. Kyushu Univ. 1978, 11,

135–143.5. Srivastava, H.M.; Saigo, M. Multiplication of fractional calculus operators and boundary value problems involving the Euler-

Darboux equation. J. Math. Anal. Appl. 1987, 121, 325–369. [CrossRef]6. Srivastava, H.M.; Agarwal, P. Certain Fractional Integral Operators and the Generalized Incomplete Hypergeometric Functions.

Appl. Appl. Math. 2013, 8, 333– 345.7. Agarwal, P.; Choi, J. Fractional calculus operators and their image formulas. J. Korean Math. Soc. 2016, 53, 1183–1210. [CrossRef]8. Virchenko, N.; Lisetska, O.; Kalla, S.L. On some fractional integral operators involving generalized Gauss hypergeometric

functions. Appl. Appl. Math. 2010, 5, 1418–1427.9. Choi, J.; Agarwal, P. A Note on Fractional Integral Operator Associated with Multiindex Mittag-Leffler Functions. Filomat 2016,

30, 1931–1939. [CrossRef]10. Agarwal, P. ; Nieto, J.J. Some fractional integral formulas for the Mittag-Leffler type function with four parameters. Open Math.

2015, 13, 537–546. [CrossRef]11. Rao, A.; Garg, M.; Kalla, S.L. Caputo-type fractional derivative of a hypergeometric integral operator. Kuwait J. Sci. Eng. 2010, 37,

15–29.12. Agarwal, P.; Choi, J.; Paris, R.B. Extended Riemann–Liouville fractional derivative operator and its applications. J. Nonlinear Sci.

Appl. 2015, 8, 451–466. [CrossRef]13. Agarwal, P.; Jain, S.; Mansour, T. Further extended Caputo fractional derivative operator and its applications. Russ. J. Math. Phys.

2017, 24, 415–425. [CrossRef]14. Kiryakova, V. On two Saigo’s fractional integral operators in the class of univalent functions. Fract. Calc. Appl. Anal. 2006, 9,

159–176.15. Kiryakova, V. A brief story about the operators of the generalized fractional calculus. Fract. Calc. Appl. Anal. 2008, 11, 203–220.16. Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations. In North-Holland

Mathematical Studies; Elsevier (North-Holland) Science Publishers: Amsterdam, The Netherlands; London, UK; New York, NY,USA, 2006; Volume 204.

17. Samko, S.G.; Kilbas, A.A.; Marichev, O.I. Fractional Integrals and Derivatives: Theory and Applications; Gordon and Breach: New York,NY, USA, 1993.

18. Sneddon, I.N. The use in mathematical physics of Erdélyi-Kober operators and of some of their generalizations. In Proceedings ofthe Fractional Calculus and Its Applications, West Haven, CT, USA, 15–16 June 1974; Ross, B., Ed.; Lecture Notes in Mathematics;Springer: Berlin/Heidelberg, Germany; New York, NY, USA, 1975; Volume 457, pp. 37–79.

19. Rainville, E.D. Special Functions; Macmillan: New York, NY, USA, 1960.20. Olver, W.J.F.; Lozier, W.D.; Boisvert, F.R.; Clark, W.C. NIST Handbook of Mathematical Functions; Cambridge University Press:

New York, NY, USA, 2010.21. Goyal, R.; Agarwal, P.; Momami, S.; Rassias, M.T. An Extension of Beta Function by Using wiman’s function. Axioms 2021, 10, 187.

[CrossRef]22. Wiman, A. U ber der Fundamentalsatz in der Theorie der Funktionen Eα(x). Acta Math. 1905, 29, 191–201. [CrossRef]23. Jain, S.; Goyal, R.; Agarwal, P.; Lupica, A.; Cesarano, C. Some results of extended beta function and hypergeometric functions by

using Wiman’s function. Mathemtics 2021, 9, 2944. [CrossRef]24. Pohlen, T. The Hadamard Product and Universal Power Series; Universitát Trier: Trier, Germany, 2009.25. Appell, P. Sur les Fonctions Hypergéométriques de Plusieurs Variables; Mémorial des Sciences Mathématiques, No. 3; Gauthier-Villars:

Paris, France, 1925.

http://doi.org/10.1016/0022-247X(87)90251-4

http://dx.doi.org/10.4134/JKMS.j150458

http://dx.doi.org/10.2298/FIL1607931C

http://dx.doi.org/10.1515/math-2015-0051

http://dx.doi.org/10.22436/jnsa.008.05.01

http://dx.doi.org/10.1134/S106192081704001X

http://dx.doi.org/10.3390/axioms10030187

http://dx.doi.org/10.1007/BF02403202

http://dx.doi.org/10.3390/math9222944

Symmetry 2022, 14, 714 15 of 15

26. Appell, P.; De Fériet, J.K. Fonctions Hypergéométriques et Hypersphériques: Polynomes d’Hermite; Gauthier-Villars: Paris, France, 1926;Volume 140.

27. Srivastava, H.M.; Karlsson, P.W. Multiple Gaussian Hypergeometric Series; Halsted Press (Ellis Horwood Limited): Chichester, UK;John Wiley and Sons: New York, NY, USA; Chichester, UK; Brisbane, Australia; Toronto, ON, Canada, 1985.

Fractional Hypergeometric Functions - MDPI

Documents

Transcript of Fractional Hypergeometric Functions - MDPI