Certain Unified Integrals Associated with Product of M-Series and Incomplete H-functions

Manish Kumar Bansal 1 ID , Devendra Kumar 2, Ilyas Khan 3,*, Jagdev Singh 4 and Kottakkaran Sooppy Nisar 5 ID 1 Department of Applied Sciences, Government Engineering College, Banswara 327001, Rajasthan, India; bansalmanish443@gmail.com 2 Department of Mathematics, University of Rajasthan, Jaipur 302004, Rajasthan, India; devendra.maths@gmail.com 3 Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City 72915, Vietnam 4 Department of Mathematics, JECRC University, Jaipur 303905, Rajasthan, India; jagdevsinghrathore@gmail.com 5 Department of Mathematics, Faculty of Arts and Sciences, Prince Sattam Bin Abdulaziz University, Wadi Al-dawasir 11991, Saudi Arabia; n.sooppy@psau.edu.sa or ksnisar1@gmail.com * Correspondence: ilyaskhan@tdtu.edu.vn


Introduction, Definitions and Preliminaries
The integral formula containing several generalized special functions (GSF) have been explored by numerous authors [1][2][3][4][5]. Many integral formulas involving GSF have been proposed and play a pivotal role in solving scientific and engineering problems. In fact, GSF are connected with different kinds of problems in various fields of mathematical sciences. These relations of GSF with different field of research have motivated many scientists to investigate the field of integrals and connected GSF. Several unified integral formulas established by many authors involving a various kind of special functions (see, for example, [6][7][8]). The key aim of this work is to develop Oberhettinger's integral formulas containing the product of M-series and incomplete H-functions. The Oberhettinger's integral formulas established in the present work are very useful to obtain the Mellin transform of various simpler special functions. The Mellin transforms of special functions find their applications in mathematical statistics, number theory, and the theory of asymptotic expansions. The main findings of the present work are very useful in solving the problems arising in digital signals, image processing, finance and ship target recognition by sonar system and radar signals [9][10][11][12]. We recall here the frequently used incomplete Gamma functions Γ( , y) and γ( , y) defined by: and respectively, satisfy the decomposition formula given by: The condition that we have used on the parameter y in and anywhere else in the current paper is unrestrained of R(z) (z ∈ C).
Srivastava et al. [13] defined incomplete generalized hypergeometric functions p Γ q and p γ q by the Mellin-Barnes type integrals in terms of incomplete Gamma functions Γ(s, y) and γ(s, y) as follows: and p γ q    (e 1 , y) , e 2 , · · · , e p ; where, L is the Mellin-Barnes type contour, having τ − i∞ as the starting point and τ + i∞ (τ ∈ R) as the end point with the usual indentations to separate a set of poles from another of the integrand in each and every case.
The incomplete H-functions γ m,n p,q (z) and Γ m,n p,q (z) have introduced and investigated by Srivastava et al. [14] (Equations (2.1)-(2.4)) in the following manner: where, . and γ m,n p,q (z) = γ m,n where, The incomplete H-functions Γ m,n p,q (z) and γ m,n p,q (z) in (6) and (7) respectively, exist for all y 0 under the same set of conditions and the same set of contour stated in the articles presented by Kilbas et al. [15], Mathai and Saxena [16] and Mathai et al. [17].
Aforementioned functions possess numerous special cases out of which we choose to enumerate a few: (i) Taking y = 0 in (6), incomplete H-function Γ m,n p,q (z) reduce to the commonly used Fox's H-function [18] as follows: (ii) Setting m = 1, n = p and replacing q by q + 1 and taking suitable parameters, the functions (6) and (7) reduces to incomplete Fox-Wright functions p Ψ (Γ) q and p Ψ (γ) q (see for details, [14] [P. 132, Equation (6.3) and (6.4)]): and (iii) Further, taking y = 0 in (9) incomplete Fox-Wright function p Ψ (Γ) q reduces to the well known Fox-Wright function p Ψ q (see for details, [18] (9) and (10), the incomplete H-functions reduces to the incomplete generalized hypergeometric functions p γ q and p Γ q (see [13]): (e 1 , y), e 2 , · · · , e p ; and (e 1 , y), e 2 , · · · , e p ; The generalized M-series was introduce and investigate by Sharma et al. [19] as follows: Here, we omit the details of convergence conditions and complete description due to all details are given in [19].

Oberhettinger's Integral Formula
Numerous author time to time establish unified integral formulas involving several kind of special function (see, for example, [21][22][23], see also, [24][25][26]). So, in this section, we establish Oberhettinger's integral formulas of the incomplete H-functions (6) and (7). The Oberhettinger's integral formulas established in the present work are very useful to obtain the Mellin transform of various simpler special functions. The Mellin transforms of special functions find their applications mathematical statistics, number theory, and the theory of asymptotic expansions. Specially, the key findings of the present study are very useful in solving the problems arising in digital signals, image processing, finance and ship target recognition by sonar system and radar signals. Theorem 1. If λ, µ ∈ C with 0 < R(µ) < R(λ) and x > 0, then the following integral formula holds: provided that the conditions of incomplete H-function Γ m,n p,q (z) in (6) are satisfied.
Proof. For the sake of convenience, let us denote the left hand side of the assertion (16) by I. Using (6) and (14) in the right side of (16) where f (ξ, y) is defined in (6). Next, upon changing the order of summation, integration and contour integral involved therein (which is permissible under the stated conditions), we obtain Finally, applying (15) in the above integral and reinterpreting it in the form of incomplete H-function Γ m,n p,q (z), we arrive at the assertion (16).
Proof. For the sake of convenience, let us denote the left hand side of the assertion (17) by J. Using (7) and (14) in the left side of (17) where F(ξ, y) is defined in (7). Next, upon changing the order of summation, integration and contour integral involved therein (which is permissible under the stated conditions), we obtain Finally, applying (15) in the above integral and reinterpreting it in the form of incomplete H-function γ m,n p,q (z), we arrive at the assertion (17).
Proof. Taking y = 0 in the result (16), we get the desired result.
Proof. Assuming (9) and (10), we get the required result here from those in Theorems 1 and 2.
Proof. Further, setting y = 0 in (19), we get the desired result.

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