A note on Bernoulli identities associated with the Weierstrass ℘-function

Abstract

Various types of Bernoulli identities have been discussed by Euler, Ramanujan, Rademacher, Eie, and others. In particular, by using certain identities involving zeta functions, Eie constructed several Bernoulli identities systematically. There are many expressions for the Bernoulli number B _2k , which express B _2k as a sum of the products of lower-order Bernoulli numbers in two terms or in three terms. In this article, we show that there is no unique method to calculate the Bernoulli number B _2k as a sum of the products of lower-order Bernoulli numbers in three terms. By employing a technique which is substantially different from the method of Eie’s proof, we derive several Bernoulli identities through the properties of the Weierstrass ℘-function. We also discuss the relationship between these and other known Bernoulli identities.

Keywords:

• Bernoulli numbers
• Weierstrass ℘-function
• Euler’s identity
• Zeta function
• Eisenstein series
• Bernoulli identities

Acknowledgements

The present investigation was supported, in part, by the Nan-Ya Institute of Technology and, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.