Eisenstein series and elliptic functions on Γ0(10)

doi:10.1016/j.aam.2009.12.007

Advances in Applied Mathematics

Volume 46, Issues 1–4, January 2011, Pages 192-208

Dedicated to Dennis Stanton on the occasion of his 60th birthday

https://doi.org/10.1016/j.aam.2009.12.007 Get rights and content

Under an Elsevier user license

open archive

Abstract

We generalize two identities involving Eisenstein series given in Chapter 19 of Ramanujanʼs second notebook. Four infinite families of Eisenstein series are obtained and their properties are investigated. The generating function of each infinite family is shown to be an elliptic function which has a simple infinite product expansion. Transformation properties of the Eisenstein series are investigated from first principles without using the transformation properties of Dedekindʼs eta-function.

MSC

primary

11F11

secondary

33E05

Keywords

Dedekindʼs eta-function

Eisenstein series

Elliptic function

Modular form

Ramanujanʼs second notebook

Theta function