Hausdorff moment sequences and hypergeometric functions

TL;DR

This paper characterizes hypergeometric functions with coefficients forming Hausdorff moment sequences, providing conditions on parameters and exploring their geometric properties, including universal starlikeness.

Contribution

It offers a complete characterization of hypergeometric functions generating Hausdorff moment sequences based on complex parameters and studies their properties.

Findings

Characterization of hypergeometric functions with Hausdorff moment sequence coefficients.

A sufficient condition for generating functions of Hausdorff moments.

Necessary and sufficient conditions for shifted hypergeometric functions to be universally starlike.

Abstract

P\'olya in 1926 showed that the hypergeometric function $F(z)=\null_2F_1(a,b;c;z)$ has a totally monotone sequence as its coefficients; that is, $F$ is the generating function of a Hausdorff moment sequence, when $0\le a\le 1$ and $0\le b\le c.$ In this paper, we give a complete characterization of such hypergeometric functions $F$ in terms of complex parameters $a,b,c.$ To this end, we study the class of general properties of generating functions of Hausdorff moment sequences and, in particular, we provide a sufficient condition for the class by making use of a Phragm\`en-Lindel\"of type theorem. As an application, we give also a necessary and sufficient condition for a shifted hypergeometric function to be universally starlike.