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