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arXiv · 1801.08670

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

D. B. Karp·J. L. López
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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.

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D. B. Karp, J. L. López. 2018-01-26. A class of Meijer's G functions and further representations of the generalized hypergeometric functions. https://arxiv.org/abs/1801.08670

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