The $p$-adic Analysis of Stirling Numbers via Higher Order Bernoulli Numbers

TL;DR

This paper explores the $p$-adic properties of Stirling numbers of both kinds using higher order Bernoulli numbers, providing simplified proofs, generalizations, and new congruences in the context of number theory.

Contribution

It introduces new $p$-adic analysis techniques for Stirling numbers, extending previous results and establishing novel congruences and proofs.

Findings

Simplified proof of $ u_2(S(2^h,k))=d_2(k)-1$ for $1 extless k extless 2^h

Generalization of $p$-adic properties to arbitrary primes $p$

New congruences modulo $p$ for Stirling numbers of both kinds

Abstract

In this paper, we use our previous study of the higher order Bernoulli numbers $B_n^{(l)}$ to investigate the $p$-adic properties of the Stirling numbers of the second kind $S(n,k)$. For example, we give a new, greatly simplified proof of the formula $\nu_2(S(2^h,k))=d_2(k)-1$ if $1\le k \le 2^h$, and generalize this result to arbitrary primes $p$. We also consider the Stirling numbers of the first kind $s(n,k)$, with new results analogous to those for the Stirling numbers of the second kind. New mod $p$ congruences for Stirling numbers of both kinds are also given.