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 $$ B_{m+n}=\sum_{\substack{0\leq k \leq n \\ 0\leq l \leq m}}\frac{(-1)^{k+l}\,k!\, l!\, S(n,k)\,S(m,l)}{(k+l+1)\,\binom{k+l}{l}}. $$ To the best of our knowledge, the identity is new.