Completely monotonic gamma ratio and infinitely divisible H-function of Fox

TL;DR

This paper establishes conditions for the complete monotonicity of gamma function ratios and explores their connection to infinitely divisible distributions via Fox's H-function, generalizing previous results for scaled cases.

Contribution

It provides necessary and sufficient conditions for monotonicity and infinite divisibility involving gamma ratios with different scaling factors, extending prior work.

Findings

Conditions for logarithmic complete monotonicity of gamma ratios.

Representation of distributions via Fox's H-function.

New integral equations for G- and H-functions.

Abstract

We investigate conditions for logarithmic complete monotonicity of a quotient of two products of gamma functions, where the argument of each gamma function has different scaling factor. We give necessary and sufficient conditions in terms of nonnegativity of some elementary function and more practical sufficient conditions in terms of parameters. Further, we study the representing measure in Bernstein's theorem for both equal and non-equal scaling factors. This leads to conditions on parameters under which Meijer's $G$-function or Fox's $H$-function represents an infinitely divisible probability distribution on positive half-line. Moreover, we present new integral equations for both $G$-function and $H$-function. The results of the paper generalize those due to Ismail (with Bustoz, Muldoon and Grinshpan) and Alzer who considered previously the case of unit scaling factors.