Fractional calculus and integral transforms of the product of a general class of polynomial and incomplete Fox–Wright functions

Motivated by a recent study on certain families of the incomplete H-functions (Srivastava et al. in Russ. J. Math. Phys. 25(1):116–138, 2018), we aim to investigate and develop several interesting properties related to product of a more general polynomial class together with incomplete Fox–Wright hypergeometric functions Ψq(γ)p(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${}_{p}\Psi _{q}^{(\gamma )}(\mathfrak{t})$\end{document} and Ψq(Γ)p(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${}_{p}\Psi _{q}^{(\Gamma )}(\mathfrak{t})$\end{document} including Marichev–Saigo–Maeda (M–S–M) fractional integral and differential operators, which contain Saigo hypergeometric, Riemann–Liouville, and Erdélyi–Kober fractional operators as particular cases regarding different parameter selection. Furthermore, we derive several integral transforms such as Jacobi, Gegenbauer (or ultraspherical), Legendre, Laplace, Mellin, Hankel, and Euler’s beta transforms.

In addition, the special functions of one or more variables are also important because they occur as solutions to these simulated differential equations. Therefore, with the development of new problems in the area of technologies in engineering and applied sciences, the subject of special functions is very diverse and is continuously growing. As a result, a number of articles on these concepts and their future implementations have been made available in the literature, see [1-4, 39, 40]. Incomplete special functions have additionally been utilized to a wide range of problems, and numerous scientific studies on incomplete special functions, along with related higher transcendental special functions, have currently been published by various authors [6-12, 15, 20-23, 31, 35-38]. In particular, the incomplete Fox-Wright functions p (γ ) q (t) and p ( ) q (t) with p numerator and q denomi-© The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. nator parameters are stated as follows [23,35]: where p q [t] is the Fox-Wright function [16]. Also, the normalized incomplete Fox-Wright functions p * (γ ) q (t) and p * ( ) q (t) are given by [23,35] p . Here, m is a positive integer (arbitrary), the coefficients A n,s ∈ R (or C) are constants (arbitrary), and (λ) ν = (λ+ν) (λ) (λ, ν ∈ C) denotes the shifted factorial (or the Pochhammer symbol). Also, the above polynomials provide a large number spectrum of well-known polynomials as one of its particular cases on appropriately specializing the coefficient A n,s . Particularly, by setting m = 1, A n,s = s! (-n) ms for s = k and A n,s = 0 for s = k, the general class of polynomials leads to a power function, i.e., Taking into account formula (1.3), it is also appropriate to study the characteristics and properties of the incomplete Fox-Wright function p ( ) q (t).

Fractional integral and differential operators
We recall a general pair of fractional integral and differential operators popularly known as Marichev-Saigo-Maeda (M-S-M), which involve, in their kernel, third Appell's twovariable hypergeometric function F 3 (.) and are defined by [32] Here, we mention and study left-hand-sided fractional integral and differential operators(see, for details, [19,26,28]).
The preceding results are well known and can be used as a proof of subsequent theorems.
Proof For the sake of simplicity, let us consider Now using (1.2) and (1.6) in (2.9) and then taking advantage of relationship (2.5), for X > 0, we acquire Finally, in opinion of the (1.2) interpretation, we get (2.8) as a desired outcome.

Integral transforms
In this part, several integral transforms such as Jacobi, Gegenbauer (or ultraspherical), Legendre, Laplace, Mellin, Hankel, and Euler's beta transforms of a product of a general polynomial class and incomplete Fox-Wright function p ( ) q (t) are presented.

Jacobi and related integral transforms
The classical orthogonal Jacobi polynomial P (h,g) n (t) is given by the following (see, for example, [33]): where 2 F 1 is the Gauss hypergeometric function [25].

Definition 2
The Jacobi transformation of a f (t) function is set as follows (see, e.g., [13, p. 501]): provided that the f (t) function seems to be so limited that only the integral exists in (3.2).
The Jacobi transform of the power function t ρ-1 is given by (see, e.g., [37, p. 128, Eq. (18)])  (21)]). In fact, this last integral formula (3.3) will be reduced instantly to the preceding form when specifying ξ = h + 1 and η = g + 1: × F 1:2;1 1:1;0 h + 1 : -n, h + g + n + 1; 1ρ; h + g + 2 : h + 1; However, in its additional limited case where ρ = m + 1 (m ∈ N 0 ), (3.4) brings the established consequence about the t m (m ∈ N 0 ) Jacobi transform studied by [25,p. 261,Eq. (14) along with (15)]: Specifying the parameters h and g, the Jacobi polynomials P (h,g) n (t) exhibit, like in their individual cases, other such recognized orthogonal polynomials being the Gegenbauer (or ultraspherical) polynomials C ν n (t), the Legendre (or spherical) polynomials P n (t), and the Tchebycheff polynomials T n (t) and U n (t) of the first kind and second kind (see, for details, [33]). In addition, we have the accompanying established connections with the Gegenbauer polynomials C ν n (t) as well as the Legendre polynomials P n (t): and P n (t) = C 1 2 respectively, which, in conjunction with (3.2), brings the Gegenbauer transform G (ν) [f (t); n] as follows: ; n ∈ N 0 , (3.8) and the resulting Legendre transform L[f (t); n] which is described by We are now generating three new results that provide the relations between Jacobi, Gegenbauer, and Legendre transforms with the following incomplete Fox-Wright function p
Proof By employing the concept of (3.2) together with (1.2), we get where, when adjusting the order of integration and summation (that might be easily explained by absolute convergence), we make use of the Jacobi transform formula (3.4) along with the parameter ρ substituted by ρ + k (ρ ∈ C; k ∈ N 0 ).
A special case of Theorem 3 when h = g = 0 (or, alternatively, Corollary 3.1 with ν = 1 2 ) gives the following result for the Legendre transform described by (3.9).
where it is assumed that the Legendre transform in (3.13) exists.

Mellin transform
The Mellin transform of a given function f (t) is represented as follows [13,29]: (3.17) given that the improper integral exists.

Hankel transform
The Hankel transform of a given function f (t) is characterized as follows [13,29]: (3.20) provided that the improper integral exists, J ν (ωt) is the Bessel function of order ν.
. (3.21) Proof Using the definition of (1.2), (1.6) and applying Hankel transform (3.20) in the lefthand side of (3.21) and then employing the formula (Prudnikov, Brychkov, and Marichev [24, (2.44) we are led easily to the right-hand side of the assertion (3.21) of Theorem 6. The details are omitted here.