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arXiv · 1407.8443

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

Wei Zhao·Jianrong Zhao·Shaofang Hong
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Abstract

Let $m, n, k$ and $c$ be positive integers. Let $ν_2(k)$ 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 $ν_2(S(c2^{n+1},2^m-1)-S(c2^n, 2^m-1))=n+1$ except when $n=m=2$ and $c=1$, in which case $ν_2(S(8,3)-S(4,3))=6$. This solves a conjecture of Lengyel proposed in 2009.

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Wei Zhao, Jianrong Zhao, Shaofang Hong. 2014-07-31. The 2-adic valuations of differences of Stirling numbers of the second kind. https://arxiv.org/abs/1407.8443

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