Eisenstein series identities involving the Borweins' cubic theta functions
Summary: Based on the theories of Ramanujan's elliptic functions and the \((p, k)\)-parametrization of theta functions due to \textit{A. Alaca} et al. [Acta Arith. 124, 177--195 (2006; Zbl 1127.11035), Int. Math. Forum 2, No. 1-4, 45--68 (2007; Zbl 1163.11003), J. Number Theory 124, No. 2, 491--510 (2007; Zbl 1169.11002)]. 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 \textit{Z.-G. Liu} [Ramanujan J. 4, No. 1, 43--50 (2000; Zbl 1012.11033)] 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. (loc. cit.).
scientific article; zbMATH DE number 6106133