Log-concavity for series in reciprocal gamma functions and applications

TL;DR

This paper studies the log-concavity properties of series involving reciprocal gamma functions, establishing conditions under which the sum retains log-concavity and applying these results to special functions like Bessel and hypergeometric functions.

Contribution

It introduces new sufficient conditions for the log-concavity of series in reciprocal gamma functions and explores their implications for special functions.

Findings

Sum is log-concave if coefficients times factorial are log-concave.

Sum is discrete Wright log-concave if coefficients are log-concave.

Derived new inequalities for special functions.

Abstract

Euler's gamma function is logarithmically convex on positive semi-axis. Additivity of logarithmic convexity implies that the function sum of gammas with non-negative coefficients is also log-convex. In this paper we investigate the series in reciprocal gamma functions, where each term is clearly log-concave. Log-concavity is not preserved by addition, so that non-negativity of the coefficients is now insufficient to draw any conclusions about the sum. We demonstrate that the sum is log-concave if the sequence of coefficients times factorial is log-concave and the sum is discrete Wright log-concave if the coefficents are log-concave. We conjecture that the latter condition is in fact sufficient for the log-concavity of the sum. We exemplify our general theorems by deriving known and new inequalities for the modified Bessel, Kummer and generalized hypergeometric functions and their parameter derivatives.