Abstract
In this paper, we obtain a ( p , q ) -extension of Srivastava’s triple hypergeometric function H B ( ⋅ ) , by using the extended Beta function B p , q ( x , y ) introduced by Choi et al. (2014). We give some of the main properties of this extended function, which include several integral representations involving Exton’s hypergeometric function, the Mellin transform, a differential formula, recursion formulas and a bounded inequality. In addition, a new integral representation of the extended Srivastava triple hypergeometric function involving Laguerre polynomials is obtained.
Highlights
Introduction and PreliminariesIn the present paper, we employ the following notations:N := {1, 2, ...}, N0 := N ∪ {0}, Z−0 := Z− ∪ {0}, where the symbols N and Z denote the set of integer and natural numbers; as usual, the symbols R and C denote the set of real and complex numbers, respectively.In the available literature, the hypergeometric series and its generalizations appear in various branches of mathematics associated with applications
The hypergeometric series and its generalizations appear in various branches of mathematics associated with applications
Our aim in this paper is to introduce a (p, q)-extension of Srivastava’s triple hypergeometric function HB(·) in (1.1), which we denote by HB,p,q(·), based on the extended Beta function in (1.6)
Summary
We employ the following notations:. N := {1, 2, ...}, N0 := N ∪ {0}, Z−0 := Z− ∪ {0}, where the symbols N and Z denote the set of integer and natural numbers; as usual, the symbols R and C denote the set of real and complex numbers, respectively. The extended Srivastava hypergeometric function HB,p,q(·) is defined in Section 2 and some integral representations are presented involving Exton’s function X4 and the Laguerre polynomials. Each of the following integral representations of the extended Srivastava triple hypergeometric function HB,p,q(·) holds for (p) > 0, (q) > 0 and min{ (b1), (b2)} > 0: Γ(b1 + b2) Γ(b1)Γ(b2). Proof : The proof of the first integral representation (2.2) follows by use of the extended beta function (1.6) in (2.1), a change in the order of integration and summation (by uniform convergence of the integral) and, after simplification, use of Exton’s triple hypergeometric function (1.3), to obtain the right-hand side of the result (2.2). The following Mellin transform of the extended Srivastava triple hypergeometric function HB,p,q(·) holds true:.
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