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arXiv · 2004.12773

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

Sumit Kumar Jha
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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.

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Sumit Kumar Jha. 2020-04-27. An identity involving Bernoulli numbers and the Stirling numbers of the second kind. https://doi.org/10.7546/nntdm.2020.26.3.160-162

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