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-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.