Recursion Formulas for Bernoulli Numbers

Abstract

In this paper we establish simple recursion formulas forBernoulli numbers, for instance,\begin{align*} \sum_{k=1}^{n}\binom{4n+2}{4k}(-1)^k 2^{2k-1}B_{4k} = n\end{align*}and\begin{align*} \sum_{k=0}^{n}\binom{4n+4}{4k+2}(-1)^k 2^{2k}B_{4k+2} = n+1\end{align*}in Theorem 1.1. Furthermore applying a Lucas sequence $V_n$,we obtain\begin{align*} \sum_{k=1}^{n}\binom{8n+4}{8k}(-1)^k 2^{2k-1}B_{8k}V_{4n-4k+2} = nV_{4n+2}\end{align*}and\begin{align*} \sum_{k=0}^{n}\binom{8n+8}{8k+4}(-1)^k 2^{2k}B_{8k+4}V_{4n-4k+2} = -(n+1)V_{4n+3}\end{align*}in Theorem 1.2.