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

Alan D. Sokal

doi:10.1016/j.jmaa.2022.126432

paper

When does a hypergeometric function $_{p} F_{q}$ belong to the Laguerre--Pólya class $L P^{+}$ ?

arXiv:2204.01045 · doi:10.1016/j.jmaa.2022.126432

Abstract

I show that a hypergeometric function $_{p} F_{q} (a_{1}, \dots, a_{p}; b_{1}, \dots, b_{q}; \cdot)$ with $p \leq q$ belongs to the Laguerre--Pólya class $L P^{+}$ for arbitrarily large $b_{p + 1}, \dots, b_{q} > 0$ 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 $_{1} F_{2}$ .

10 pages, LaTeX2e

When does a hypergeometric function p​Fq​ belong to the Laguerre--Pólya class LP+?

Abstract

References in corpus (3)

When does a hypergeometric function $_{p} F_{q}$ belong to the Laguerre--Pólya class $L P^{+}$ ?