  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.
