An identity involving Bernoulli numbers and the Stirling numbers of the second kind

Abstract

Let $B_n$ denote the Bernoulli numbers, and $S(n,k)$ denote the Stirling numbers of the second kind. We prove the following identity $$B_{m+n}=\sum _{\begin{array}{c}0\le k\le n\\ 0\le l\le m\end{array}}\frac{(-1{)}^{k+l}k!l!S(n,k)S(m,l)}{(k+l+1)\left(\genfrac{}{}{0ex}{}{k+l}{l}\right)}.$$ To the best of our knowledge, the identity is new.

3 pages