New results on the p-adic valuation of Stirling numbers

TL;DR

This paper advances the understanding of the p-adic valuations of Stirling numbers of both kinds, providing new estimates, explicit evaluations, and criteria for sharpness, with broad generalizations across all primes.

Contribution

It introduces new bounds, explicit formulas, and criteria for p-adic valuations of Stirling numbers, extending previous results and proposing new conjectures.

Findings

Explicit evaluation of (S(n,k)) for specific n and k

New estimates for (S(n,k)) and (s(n,k))

Proved new identities and extended previous results

Abstract

We generalize results on the $p$-adic valuations of $S(n,k)$, the Stirling number of the second kind and $s(n,k)$ the Stirling number of the first kind. We have several new estimates for these valuations, along with criteria for when the estimates are sharp. The primary foci are the explicit evaluation of $\nu_2(S(n,k))$ with $n=c2^h$, $k=b2^h+a$, $a, b, c, h, k, n \in Z^+$, and $1\le a \le 2^{h-1}$, and $\nu_p(S(n,k))$ when $n=cp^h$ for an odd prime $p$. We have strong new results, which generalize and strengthen previous results, for all primes. We also have some new results on the $p$-adic valuations $\nu_p(s(n,k))$ for all primes. We generally assume that $p-1|n-k$ for exact values of $\nu_p(S(n,k))$ or $\nu_p(s(n,k))$. In addition, we have proved some new Amdeberhan-type identities for Stirling numbers of both kinds. We also extend some recent results and propose two new conjectures, as well as proofs and extensions of previous ones.