When does a hypergeometric function ${}_p{F}_q$ belong to the Laguerre--Pólya class $L{P}^{+}$?

Abstract

I show that a hypergeometric function ${}_p{F}_q(a_1,\dots ,a_p;b_1,\dots ,b_q;\cdot )$ with $p\le q$ 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 ${}_1{F}_2$.

arXiv comment10 pages, LaTeX2e

References in corpus (6)

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• Newton diagram of positivity for ${}_1{F}_2$ generalized hypergeometric functions
• Multiple Laguerre polynomials: Combinatorial model and Stieltjes moment representation