The 2-adic valuations of differences of Stirling numbers of the second kind

TL;DR

This paper analyzes the 2-adic valuations of differences of Stirling numbers of the second kind, establishing a convolution identity and proving a conjecture about their valuations with detailed 2-adic analysis.

Contribution

It introduces a new convolution identity for Stirling numbers of the second kind and proves a conjecture regarding their 2-adic valuations, advancing understanding of their number-theoretic properties.

Findings

Established a convolution identity for Stirling numbers of the second kind.

Proved the 2-adic valuation formula for differences of these numbers under certain conditions.

Solved Lengyel's 2009 conjecture on 2-adic valuations.

Abstract

Let $m, n, k$ and $c$ be positive integers. Let $\nu_2(k)$ be the 2-adic valuation of $k$. By $S(n,k)$ we denote the Stirling numbers of the second kind. In this paper, we first establish a convolution identity of the Stirling numbers of the second kind and provide a detailed 2-adic analysis to the Stirling numbers of the second kind. Consequently, we show that if $2\le m\le n$ and $c$ is odd, then $\nu_2(S(c2^{n+1},2^m-1)-S(c2^n, 2^m-1))=n+1$ except when $n=m=2$ and $c=1$, in which case $\nu_2(S(8,3)-S(4,3))=6$. This solves a conjecture of Lengyel proposed in 2009.