Derangements and the $p$-adic incomplete gamma function

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.

26 pages; new material on m-values; revised for clarity

References in corpus (4)

• Euler's constant: Euler's work and modern developments
• Searching for a counterexample to Kurepa's Conjecture
• Congruences on the Bell polynomials and the derangement polynomials
• Functions with integer-valued divided differences