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arXiv · 2207.11806

P-adic incomplete gamma functions and Artin-Hasse-type series

Xiaojian Li·Jay Reiter·Shiang Tang·Napoleon Wang·Jin Yi
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Abstract

We define and study a $p$-adic analogue of the incomplete gamma function related to Morita's $p$-adic gamma function. We also discuss a combinatorial identity related to the Artin-Hasse series, which is a special case of the exponential principle in combinatorics. From this we deduce a curious $p$-adic property of $|\mathrm{Hom} (G,S_n)|$ for a topologically finitely generated group $G$, using a characterization of $p$-adic continuity for certain functions $f \colon \mathbb Z_{>0} \to \mathbb Q_p$ due to O'Desky-Richman. In the end, we give an exposition of some standard properties of the Artin-Hasse series.

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Xiaojian Li, Jay Reiter, Shiang Tang, Napoleon Wang, Jin Yi. 2022-07-24. P-adic incomplete gamma functions and Artin-Hasse-type series. https://arxiv.org/abs/2207.11806

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