Derangements and the 𝑝-adic incomplete gamma function

Abstract

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