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

TL;DR

This paper derives an inverse factorial series expansion for gamma function ratios with linear arguments, explores properties of N{ o}rlund-Bernoulli polynomials, and applies these findings to analyze Fox's H function near singularities.

Contribution

It introduces a new inverse factorial series expansion for gamma ratios, relates coefficients to N{ o}rlund-Bernoulli polynomials, and provides insights into the convergence and applications to special functions.

Findings

Derived an inverse factorial series expansion for gamma ratios.

Established recurrence relations involving N{ o}rlund-Bernoulli polynomials.

Applied the expansion to analyze Fox's H function near singular points.

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.