Some new facts concerning the delta neutral H function of Fox

TL;DR

This paper explores new properties of delta neutral Fox H functions, including expansions, Mellin transforms, and applications to measure representation and integral equations, advancing understanding of these special functions.

Contribution

It introduces new expansions, Mellin transform formulas, and applications to measure representation and integral equations for delta neutral Fox H functions.

Findings

Derived expansions near finite nonzero singularities.

Proved a conjecture on the representing measure for gamma ratios.

Extended integral equations to the zero-balanced case.

Abstract

In this paper we find several new properties of a class of Fox's H functions which we call delta neutral. In particular, we find an 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 the weak limit of measures expressed in terms of the H function which furnishes a regularization method for integrals containing the delta neutral zero-balanced function of Fox. We apply this result to extend a recently discovered integral equation to zero-balanced case. In the last section of the paper we consider a reduced form of this integral equation for Meijer's G function. This leads to certain expansions believed to be new even in the case of the Gauss hypergeometric function.