Partial fraction expansions and zeros of Hankel transforms

TL;DR

This paper proves that certain Hankel transforms have all real zeros distributed regularly, and investigates conditions for hypergeometric functions to belong to the Laguerre-Pólya class, using partial fractions and Sturm's theory.

Contribution

It introduces a novel approach combining partial fraction expansions and Sturm's oscillation theory to analyze zeros of Hankel transforms and hypergeometric functions.

Findings

Zeros of certain Hankel transforms are all real and regularly distributed.

Conditions for hypergeometric functions to be in the Laguerre-Pólya class are established.

Method provides a constructive way to analyze zeros of special functions.

Abstract

It is proved by the method of partial fraction expansions and Sturm's oscillation theory that the zeros of certain Hankel transforms are all real and distributed regularly between consecutive zeros of Bessel functions. As an application, the sufficient or necessary conditions on parameters for which ${}_1F_2$ hypergeometric functions belong to the Laguerre-P\'olya class are investigated in a constructive manner.