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

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$c$isodd,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.