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arXiv · 2204.01045

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

Alan D. Sokal
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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 ${}_{1}F_2$.

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Alan D. Sokal. 2022-04-03. When does a hypergeometric function ${}_{p\!}F_q$ belong to the Laguerre--P\'olya class $LP^+$?. https://doi.org/10.1016/j.jmaa.2022.126432

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