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Hausdorff moment sequences and hypergeometric functions

Toshiyuki Sugawa, Li-Mei Wang

Abstract

Pólya 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 Phragmèn-Lindelöf type theorem. As an application, we give also a necessary and sufficient condition for a shifted hypergeometric function to be universally starlike.

Hausdorff moment sequences and hypergeometric functions

Abstract

Pólya in 1926 showed that the hypergeometric function has a totally monotone sequence as its coefficients; that is, is the generating function of a Hausdorff moment sequence, when and In this paper, we give a complete characterization of such hypergeometric functions in terms of complex parameters 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 Phragmèn-Lindelöf type theorem. As an application, we give also a necessary and sufficient condition for a shifted hypergeometric function to be universally starlike.

Paper Structure

This paper contains 5 sections, 14 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Proof of the main results
  5. Application to universally starlike functions

Key Result

Lemma 2.1

Let $c_0=1,c_1,c_2,\dots$ be a totally monotone sequence with representing probability measure $\mu.$ Then the following assertions hold.

Theorems & Definitions (14)

  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Theorem 2.4: Phragmèn-Lindelöf principle
  • Lemma 2.5
  • Theorem 2.6
  • Lemma 2.7
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3
  • ...and 4 more