An inverse factorial series for a general gamma ratio and related properties of the N{\o}rlund-Bernoulli polynomials

Abstract

We find an inverse factorial series expansion for the ratio of products of gamma functions whose arguments are linear functions of the variable. We a give recurrence relation for the coefficients in terms of the N{\o}rlund-Bernoulli polynomials and determine quite precisely the half-plane of convergence. Our results complement naturally a number of previous investigations of the gamma ratios which began in the 1930ies. The expansion obtained in this paper plays a crucial role in the study of the behavior of the delta-neutral Fox's H function in the neighborhood of it's finite singular point. We further apply a particular case of the inverse factorial series expansion to derive a possibly new identity for the N{\o}rlund-Bernoulli polynomials.