A class of Meijer's G functions and further representations of the generalized hypergeometric functions

Abstract

In this paper we investigate the Meijer's G function G p,1 p+1,p+1 which for certain parameter values represents the Riemann-Liouville fractional integral of Meijer-Nørlund function G p,0 p,p . Our results for G p,1 p+1,p+1 include: a regularization formula for overlapping poles, a connection formula with the Meijer-Nørlund function, asymptotic formulas around the origin and unity, formulas for the moments, a hypergeometric transform and a sign stabilization theorem for growing parameters. We further employ the properties of G p,1 p+1,p+1 to calculate the Hadamard finite part of an integral containing the Meijer-Nørlund function that is singular at unity. In the ultimate section, we define an alternative regularization for such integral better suited for representing the Bessel type generalized hypergeometric function p−1 F p . A particular case of this regularization is then used to identify some new facts about the positivity and reality of the zeros of this function.

The Transmutation Method and Boundary-Value Problems for Singular Elliptic Equations

“…Moreover, once the circuit over the region with poles is over, the contour is continued up and down along the same vertical line. Is another contour is selected (such that the infinity is achieved along horizontal lines), then new modifications of Meyer functions are obtained (see [521,522,525]). In such cases, it is possible that the Meyer function is defined, but its Mellin transform dos not exist, or it exists, but the function itself is not restored by means of the inverse Mellin transformation (via the Mellin-Barnes integral).…”

“…A special case of the Meyer function G p,p p,0 important for applications is comprehensively investigated by Nørlund; this is the reason to propose the term Meyer-Nørlund function (see [521,522,525]).…”