Derangements and the $p$-adic incomplete gamma function

Abstract

We introduce a new p-adic analogue of the incomplete gamma function. We also introduce a closely related family of combinatorial sequences counting derangements and arrangements in certain wreath products.This means expressions outside of the classical range of definition can be computed relative to a given prime. For instance, for any prime p, the limit d −1 := lim n→∞ d p n −1 (in the p-adic topology)

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

“…Morita's p-adic gamma function [5] is a p-adic analogue of the classical gamma function that has been extensively studied in the literature. On the other hand, recent work of O'Desky-Richman [6] defines and studies a p-adic analogue of the incomplete gamma function Γ(s, z) := ∞ z t s−1 e −t dt defined for certain complex numbers s, z. However, it is unclear how their gamma function may be related to Morita's gamma function.…”

“…(Note that their term "p-adically continuous" is equivalent to "extendable to a continuous p-adic function on Z p ". )The special case when G = Z p is[6, Corollary 1.3].…”