On Eisenstein series and ∑_{𝑚,𝑛=-∞}^{∞}𝑞^{𝑚²+𝑚𝑛+2𝑛²}

Abstract

In this paper, we derive some new identities satisfied by the series ∑ m , n = − ∞ ∞ q m 2 + m n + 2 n 2 {\sum _{m,n=-\infty }^{\infty }q^{m^{2}+mn+2n^{2}}} using Ramanujan’s identities for L ( q ) L(q) , M ( q ) M(q) and N ( q ) N(q) . Our work is motivated by an attempt to develop a theory of elliptic functions to the septic base.