Journal of Applied Mathematics
Volume 2012 (2012), Article ID 181264, 14 pages
Research Article

Eisenstein Series Identities Involving the Borweins' Cubic Theta Functions

Department of Mathematics, Jiangsu University, Jiangsu, Zhenjiang 212013, China

Received 28 March 2012; Revised 15 May 2012; Accepted 23 May 2012

Academic Editor: Ferenc Hartung

Copyright © 2012 Ernest X. W. Xia and Ferenc Hartung. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Based on the theories of Ramanujan's elliptic functions and the (p, k)-parametrization of theta functions due to Alaca et al. (2006, 2007, 2006) we derive certain Eisenstein series identities involving the Borweins' cubic theta functions with the help of the computer. Some of these identities were proved by Liu based on the fundamental theory of elliptic functions and some of them may be new. One side of each identity involves Eisenstein series, the other products of the Borweins' cubic theta functions. As applications, we evaluate some convolution sums. These evaluations are different from the formulas given by Alaca et al.