Applications of a recurrence for the Bernoulli numbers
type: Item
instance of: scholarly article
title: Applications of a recurrence for the Bernoulli numbers
zbMATH Open document ID: 0844.11019
author: Fredric T. Howard
published in: Journal of Number Theory
publication date: 1995-06-19T00:00:00Z
review text: The author provides an easy proof of the recurrence \[ B_m= {1\over {n(1- n^m)}} \sum^{m-1}_{k =0} n^k {m \choose k} B_k \sum^{n-1}_{j=1} j^{m-k}, \] where \(\{B_m\}\) are the Bernoulli numbers. The author uses this formula to present proofs of theorems on Bernoulli numbers due to Staudt-Clausen, Carlitz, Frobenius and Ramanujan. An analogous recurrence for Genocchi numbers is given, which is used to present new proofs of theorems on Genocchi numbers of Lehmer, Ramanujan and Kummer. In some cases, earlier results are extended.
Mathematics Subject Classification ID: 11B68
zbMATH DE Number: 765168
zbMATH Keywords: recurrence
zbMATH Keywords: Bernoulli numbers
zbMATH Keywords: Genocchi numbers
reviewed by: Matthew F. Wyneken
MaRDI profile type: Publication
full work available at URL: jnth.1995.1062
OpenAlex ID: W2083424528
DOI: 10.1006/JNTH.1995.1062
description: scientific article; zbMATH DE number 765168