Applications of a Recurrence for the Bernoulli Numbers

Abstract

We give an easy proof of a recently published recurrence for the Bernoulli numbers and we present some applications of the recurrence. One of the applications is a simple proof of the well-known Staudt-Clausen Theorem. Proofs are also given for theorems of Carlitz, Frobenius, and Ramanujan. An analogous recurrence for Genocchi numbers is proved and applications are given. In particular, theorems of Lehmer, Ramanujan, and Kummer are proved and, in some cases, extended.