On Meijer's $G$ function $G^{m,n}_{p,p}$ for $m+n=p$

Abstract

The paper is devoted to the piece-wise analytic case of Meijer's $G$ function $G^{m,n}_{p,p}$. While the problem of its analytic continuation was solved in principle by Meijer and Braaksma we show that in the ''balanced'' case $m+n=p$ the formulas take a particularly simple form. We derive explicit expressions for the values of these analytic continuations on the banks of the branch cuts. It is further demonstrated that particular cases of this type of $G$ function having integer parameter differences satisfy identities similar to the Miller-Paris transformations for the generalized hypergeometric function. Finally, we give a presumably new integral evaluation involving $G^{m,n}_{p,p}$ function with $m=n$ and apply it for summing a series involving digamma function and related to the power series coefficients of the product of two generalized hypergeometric functions with shifted parameters.