Abstract
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.
Highlights
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
We evaluate some convolution sums
In this paper, using the parameters p and k introduced by Alaca et al 9–11 see 2.1, we deduce some Eisenstein series identities involving the Borweins’ cubic theta functions with the help of the computer
Summary
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. One side of each identity involves Eisenstein series, the other products of the Borweins’ cubic theta functions.
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