Derangements and the $p$-adic incomplete gamma function

TL;DR

This paper introduces a $p$-adic analogue of the incomplete gamma function, explores its properties, and provides combinatorial interpretations related to derangements and restricted permutations.

Contribution

It develops a new $p$-adic incomplete gamma function, characterizes $p$-adic continuity via $m$-values, and links these functions to combinatorial counting problems.

Findings

$p$-adic incomplete gamma function defined and analyzed.

Characterization of $p$-adic continuity using $m$-values.

Combinatorial interpretations involving derangements and restricted permutations.

Abstract

We introduce a $p$-adic analogue of the incomplete gamma function. We also introduce quantities ($m$-values) associated to a function on natural numbers and prove a new characterization of $p$-adic continuity for functions with $p$-integral $m$-values. Combinatorial interpretations for the integral values of the incomplete gamma function and functions with $m$-values zero or one are obtained, which show that these functions count derangements in generalized symmetric groups and permutations with restricted cycle lengths.