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 # Hom(G, Sn) for a topologically finitely generated group G, using a characterization of p-adic continuity for certain functions f : Z>0 → Qp due to O'Desky-Richman. In the end, we give an exposition of some standard properties of the Artin-Hasse series.