Some expansion formulas for incomplete H- and H̅-functions involving Bessel functions

In this paper, we assess an integral containing incomplete H-functions and utilize it to build up an expansion formula for the incomplete H-functions including the Bessel function. Next, we evaluate an integral containing incomplete H̅-functions and use it to develop an expansion formula for the incomplete H̅-functions including the Bessel function. The outcomes introduced in this paper are general in nature, and several particular cases can be acquired by giving specific values to the parameters engaged with the principle results. As particular cases, we derive expansions for the incomplete Meijer G(Γ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${}^{(\Gamma )}G$\end{document}-function, Fox–Wright Ψq(Γ)p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${}_{p}\Psi _{q}^{(\Gamma )}$\end{document}-function, and generalized hypergeometric Γqp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${}_{p}\Gamma _{q}$\end{document} function.


Introduction and preliminaries
The topic of special functions is very rich and is constantly increasing with the advent of new problems in the field of applications in engineering and applied sciences. In addition, applications to H-functions was already documented in a large range of response-related topics, such as diffusion, reaction-diffusion, electronics and communication, fractional differential and additive equations, other fields of theoretical physics, biology, and mathematical probability theory. Therefore, due to the overwhelming demand, a number of papers on these functions and their possible applications have been made available in the literature. For further information, see the research monographs [18,19] and recent work [1][2][3]16]. The main object of this paper is to build up integrals including incomplete Hfunctions and incomplete H-functions and utilize them to get expansions for incomplete Fox [12] investigated and defined a new function during his study of symmetrical Fourier kernels in terms of the Mellin-Barnes-type contour integral, known as Fox's H-function H m,n p,q (z) = H m,n p,q z (a j , A j ) 1,p where L is a convenient contour that detached the poles. m, n, p, q are positive integers with constraints 0 ≤ n ≤ p, 1 ≤ m ≤ q, the coefficients A j (j = 1, . . . , p) and B j (j = 1, . . . , q) ∈ R + , and a j and b j are complex parameters. The H-function is absolutely convergent and defines an analytic function under the set of conditions described in [12] (see also [15,18,19]). In 1987, Inayat-Hussain [13] introduced a generalization of the H-functions, known as the H-functions: where m, n, p, q ∈ N 0 with constraints 0 ≤ n ≤ p, 1 ≤ m ≤ q, A j (j = 1, . . . , p), B j (j = 1, . . . , q) ∈ R + , a j (j = 1, . . . , p), and b j (j = 1, . . . , q) are complex numbers. The exponents ζ j (j = 1, . . . , n) and η j (j = m + 1, . . . , q) take noninteger values, and L is a reasonable contour that detaches the poles. The H-function is absolutely convergent under the arrangement of conditions described by Buschman and Srivastava [10]. We next recollect and define the lower and upper incomplete gamma functions γ (ϑ, z) and (ϑ, z) as follows: and These functions fulfill the following relation: Using the incomplete gamma functions defined above, Srivastava et al. [23] presented and researched the incomplete H-functions as follows: and m,n where and with the arrangement of conditions setout in [23]. These incomplete H-functions fulfill the following relation (known as decomposition formula): The incomplete H-functions γ m,n p,q (z) and m,n p,q (z) defined in (8) and (9) exist for x ≥ 0 under the set of conditions given by Srivastava et al. [23], with Srivastava et al. [23] introduced a generalization of the incomplete H-functions, known as the incomplete H-functions defined in the following way: and m,n p,q (z) = m,n p,q z (a 1 , A 1 ; ζ 1 ; y), (a j , A j ; ζ j ) 2,n , (a j , where and with the arrangements of conditions setout in [23] with Several authors currently work on a wide variety of applications for these incomplete functions. See, for example, recent works [6-9, 14, 20-22] and references therein.
The paper is organized in the following way. In Sect. 2, we evaluate the improper integrals involving the Bessel function, incomplete H-functions, and incomplete H-functions. In Sect. 3, we derive expansions for incomplete H-functions and incomplete H-functions involving the Bessel function with the help of integrals presented in Sect. 2 and the orthogonal properties of Bessel functions. In Sect. 4, we obtain particular cases.

The integrals
In this section, we derive improper integrals involving the Bessel function, incomplete Hfunctions, and incomplete H-functions. These integrals will be used in Sect. 3 to prove the expansions for incomplete H-functions and incomplete H-functions.
Proof To demonstrate (18), consider its the left-hand side. Expressing the incomplete Hfunction in terms of the Mellin-Barnes-type integral defined in (9), we have Change the integration order, we have To assess the above internal integral, we will use the following formula [17, p. 106, (1)]: Then we get Using (11), we obtain the required right-hand side of (18).
Proof To demonstrate (22), consider its left-hand side. Expressing the incomplete Hfunction in terms of the Mellin-Barnes-type integral defined in (14), we have Changing the integration order, we have From this by means of formula (19) we obtain and using (16), we obtain the required right-hand side of (22).

Expansion formulas
In this section, we present expansions for incomplete H-functions and incomplete Hfunctions involving the Bessel function with the help of integrals presented in Sect. 2 and derive the orthogonality of Bessel functions.

Particular cases
The results presented in this paper are of a very general nature, and their particular cases are scattered throughout the literature. Particular cases of expansion are mentioned only for the incomplete m,n p,q function.
If we assign specific values to the parameters of the incomplete m,n p,q function, then this function converts into the incomplete Meijer ( ) G-function, incomplete Fox-Wright p ( ) q -function, and incomplete generalized hypergeometric p q function. In this Section, we establish integral formulas and expansion formulas for these incomplete functions as particular cases of Theorem 2.1 and Theorem 3.1.