Certain Properties of Generalized M-Series under Generalized Fractional Integral Operators

)e aim of this study is to introduce new (presumed) generalized fractional integral operators involving I-function as a kernel. In addition, two theorems have been developed under these operators that provide an image formula for this generalized M-series and also to study the different properties of the generalized M-series. )e corresponding assertions in terms of Euler and Laplace transform methods are presented. Due to the general nature of the I-function and the generalized M-series, a number of results involving special functions can be achieved only by making appropriate values for the parameters.

e generalized M-series is important because its basic cases are followed by the Mittag-Leffler function and hypergeometric function, and all these functions have actually discovered key implementations in solving problems in applied sciences, chemistry, physics, and biology. A number of researchers [6][7][8][9][10] have also investigated the structure, implementations, and various directions of extensions of the fractional integration and differentiation in detail. e series is defined for z, φ, ς ∈ C, R(φ) > 0, and α i , β j ∈ R(−∞, ∞), (α i : i � 1, 2, . . . , p; β j ≠ 0: j � 1, 2, . . . , q) as where (α j ) k , (β j ) k are showing the results for Pochhammer symbols. e series (1) is defined when none of the parameters β j s, (j � 1, . . . , q) is a negative integer or zero; if any numerator parameter α j is a negative integer or zero, then the series terminates to a polynomial in z. e series in (1) is convergent for all z if p ≤ q, it is convergent for |z| < ϑ � φ φ if , and it is divergent if p > q + 1. When p � q + 1 and |z| < ϑ, the series can converge on conditions depending on the parameters ( [2], for the general theory of the Wright function). e summation of the convergent series is denoted by the symbol p M φ,ς q (.). Some essential special cases of the generalized M-series are mentioned in the following: (1) For φ � ς � 1, the generalized M-series is the generalized hypergeometric function [11,12].
the generalized M-series reduces to the generalized Mittag-Leffler function [12,15] as follows: In the present study, our aim is to study some fundamental properties of generalized M-series defined by (1), for which, we consider the two generalized fractional integral operators involving the I-function as kernel, which is described in the next section.

Generalized Fractional Integral Operators
In this section, we are introducing new (presumed) generalized fractional integral operators involving I-function as kernel, which are the extensions of Saxena and Kumbhat operators [16,17]: and where e sufficient conditions of these operators are where the I-function, which is more general than Fox's H-function, is defined by Saxena [18], by means of the following Mellin-Barnes type contour integral: where ω � �� � −1 √ and 2 Journal of Mathematics p i , q i (i � 1, . . . , r), m, n are the integers satisfying 0 ≤ n ≤ p i , 0 ≤ m ≤ q i ; A j , B j , A ji , B ji are the real and positive numbers, and a j , b j , a ji , b ji are the complex numbers. L is a suitable contour of the Mellin-Barnes type running from c − iφ to c + iφ (c is real) in the complex ζ-plane. Details regarding existence conditions and various parametric restrictions of I-function are provided by Saxena [18].

Images of Generalized M-Series under the Generalized Fractional Integral Operators
In this section, we established the image formula for the generalized M-series (1) under the generalized fractional integral operators (6) and (7) in terms of the I-function as the kernel. e results are given in eorems 1 and 2.
provided the conditions, stated with operator (6), are satisfied.
Proof. We assume Ω 1 be the on the left-hand side of (14); using the definition of generalized M-series (1) and the generalized fractional integral operator (6) on the left-hand side of (14), we have Now, by changing the order of the integration which is valid under the given with theorem, we get Let the substitution t r /x r � w and then t � xw 1/r in (16), we get Journal of Mathematics Using the definition of the well-known beta function in the inner integral, we have Interpreting the right-hand side of (18), in view of the definition (11), we arrive at the result (14).
provided the conditions, stated with operator (7), are satisfied. Proof. On the left-hand side of (19), let Ω 2 , using (1) and (7) on the left-hand side of (19), we have Now, by changing the order of the integration which is valid under the given stated theorem, we get Let the replacement x r /t r � w and then t � x/w 1/r in (21), we get By beta function, we have 4 Journal of Mathematics Interpreting the right-hand side of (23), in view of the definition (11), we arrive at the result (19).

Special Cases
(1) If we put φ � ς � 1 in eorems 1 and 2, we obtain the following interesting results on the right, and it is known as the generalized hypergeometric function.

Journal of Mathematics
(2) If we put p � q � 0 in eorems 1 and 2, we obtain the following interesting results on the right, and it is known as the two-index Mittag-Leffler function.

Certain Integral Transforms
In this section, with the aid of the results developed in the prior segment, we will provide some very important outcomes of several theorems connected with the transforms of Euler and Laplace. To this end, we would like to define these transforms first.
Definition 1. e well-known Euler transform (e.g., [19]) of a function f(t) is defined as Definition 2. e Laplace transform (e.g., [19]) of the function f(t) is defined, as usual, by is section would establish the following fascinating outcomes in the form of theorems. As these findings are direct implications of Definitions 1 and 2 and eorems 1 and 2, they are provided without evidence here.

Theorem 3.
e Euler transform of the eorem 1 gives the following result: p+2,q+1;r λ| a j , A j 1,n ; a ji , A ji n+1, provided that the conditions mentioned with the operator and Euler transform are satisfied.

Theorem 4.
e Euler transform of the eorem 2 gives the following result: p+2,q+1;r λ| a j , A j 1,n ; a ji , A ji n+1, provided that the conditions mentioned with the operator and Laplace transform are satisfied.

Theorem 6.
e Laplace transform of the eorem 2 gives the following result: p+2,q+1;r λ| a j , A j 1,n ; a ji , A ji n+1, provided that the conditions mentioned with the operator and Laplace transform are satisfied.