Some new facts concerning the delta neutral H function of Fox

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.