When does a hypergeometric function ${}_{p\!}F_q$ belong to the Laguerre--P\'olya class $LP^+$?

TL;DR

This paper characterizes when hypergeometric functions ${}_{p}F_q$ belong to the Laguerre--Pólya class $LP^+$, showing it depends on the differences of parameters being nonnegative integers, with explicit examples provided.

Contribution

It establishes a precise criterion for hypergeometric functions ${}_{p}F_q$ to be in $LP^+$ based on parameter differences, extending previous results for the case $p=q$.

Findings

Hypergeometric functions ${}_{p}F_q$ belong to $LP^+$ if parameter differences are nonnegative integers.

The result generalizes earlier work for the case $p=q$ by Ki and Kim.

Explicit examples for ${}_{1}F_2$ illustrate the criterion.

Abstract

I show that a hypergeometric function ${}_{p}F_q(a_1,\ldots,a_p;b_1,\ldots,b_q;\,\cdot\,)$ with $p \le q$ belongs to the Laguerre--P\'olya class $LP^+$ for arbitrarily large $b_{p+1},\ldots,b_q > 0$ if and only if, after a possible reordering, the differences $a_i - b_i$ are nonnegative integers. This result arises as an easy corollary of the case $p=q$ proven two decades ago by Ki and Kim. I also give explicit examples for the case ${}_{1}F_2$.