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

TL;DR

This paper explores a specific class of Meijer's G functions, deriving new formulas, asymptotic behaviors, and regularizations, which enhance understanding of their properties and applications to hypergeometric functions.

Contribution

It introduces new regularization formulas, connection formulas, and asymptotic results for the Meijer-G function $G^{p,1}_{p+1,p+1}$, expanding its theoretical framework and applications.

Findings

Derived a regularization formula for overlapping poles.

Established a connection with Meijer-N{ }rlund functions.

Provided asymptotic formulas and positivity results.

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-N{\o}rlund 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-N{\o}rlund 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-N{\o}rlund 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.