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

2017/07/06 by Karp, Dmitrii B., Prilepkina, Elena G.
#11B68 #30B50 #33B15 #41A58 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1707.01734

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 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.

Related

• The asymptotic expansion for the factorial and Lagrange inversion formula
• Bounds for the logarithm of the Euler gamma function and its derivatives
• Stirling numbers and inverse factorial series
• An extension of the Bernoulli polynomials inspired by the Tsallis statistics
• Asymptotic expansions for ratios of products of gamma functions