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

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 $\mathrm{\mid }\mathrm{H}\mathrm{o}\mathrm{m}(G,S_n)\mathrm{\mid }$ for a topologically finitely generated group $G$, using a characterization of $p$-adic continuity for certain functions $f\text{\hspace{0.17em}⁣}:{\mathbb{Z}}_{>0}\to {\mathbb{Q}}_p$ due to O’Desky-Richman. In the end, we give an exposition of some standard properties of the Artin-Hasse series.