p-adic Stirling numbers of the second kind

TL;DR

This paper investigates the p-adic limits of Stirling numbers of the second kind, establishing their existence and expressing them in terms of p-adic binomial coefficients under certain conditions.

Contribution

It proves the existence of p-adic limits of Stirling numbers of the second kind and provides explicit formulas in terms of p-adic binomial coefficients.

Findings

p-adic limits of S(n,k) exist for all integers a, b, c, d

Explicit expressions for limits when a ≡ b mod (p-1) or d ≤ 0

Introduction of p-adic Stirling numbers as limits

Abstract

Let S(n,k) denote the Stirling numbers of the second kind. We prove that the p-adic limit of S(p^e a + c, p^e b + d) as e goes to infinity exists for all integers a, b, c, and d. We call the limiting p-adic integer S(p^\infty a + c, p^\infty b + d). When a equiv b mod (p-1) or d \le 0, we express them in terms of p-adic binomial coefficients introduced in a recent paper.