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 $G_{p,p}^{p,0}$. 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.

23 pages, no figures

References in corpus (2)

• Inequalities and monotonicity of ratios for generalized hypergeometric function
• Newton diagram of positivity for ${}_1{F}_2$ generalized hypergeometric functions