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

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 $f:{\mathbb{Z}}_{>0}\to {\mathbb{Q}}_p$ due to O'Desky-Richman. In the end, we give an exposition of some standard properties of the Artin-Hasse series.

Replaced the incorrect Prop 2.8 by Remark 2.8. To appear in p-Adic Numbers, Ultrametric Analysis and Applications