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

TL;DR

This paper introduces a $p$-adic incomplete gamma function, explores its relation to Artin-Hasse series, and uncovers new $p$-adic properties of group homomorphism counts, connecting combinatorics and number theory.

Contribution

It defines a novel $p$-adic incomplete gamma function, establishes a combinatorial identity involving Artin-Hasse series, and reveals new $p$-adic properties of homomorphism counts for finitely generated groups.

Findings

A new $p$-adic incomplete gamma function is defined.

A combinatorial identity related to Artin-Hasse series is established.

A $p$-adic property of $| ext{Hom}(G,S_n)|$ is deduced.

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.