An inverse factorial series for a general gamma ratio and related properties of the Nørlund–Bernoulli polynomials

We find an inverse factorial series expansion for the ratio of products of gamma functions whose arguments are linear functions of the variable. We give a recurrence relation for the coefficients in terms of the Nørlund–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 Nørlund–Bernoulli polynomials. Bibliography: $49$ titles.