Some new facts around the delta neutral H function of Fox

D. Karp, E. Prilepkina

Published 2015-11-20Version 1

In this paper we find several new properties of delta neutral H function of Fox. In particular, we find expansion in the neighborhood of finite nonzero singularity and give new Mellin transform formulas under a special restriction on parameters. The last result is applied to prove a conjecture regarding the representing measure for gamma ratio in Bernstein's theorem. Further, we find weak limit of measures expressed in terms of H function providing a regularization method for integrals containing delta neutral zero-balanced H function. We apply this result to extend a recently discovered integral equation to zero-balanced case. In the last section we consider this integral equation in case of Meijer's G function which leads to certain expansions believed to be new even in the case of the Gauss hypergeometric function.