Some Eisenstein series identities related to modular equations of the seventh order

In this paper we will use one well-known modular equation of seventh order, one theta function identity of S. McCullough and L.-C. Shen, 1994, and the complex variable theory of el-liptic functions to prove some new septic identities for theta functions. Then we use these identities to provide new proofs of some Eisenstein series identities in Ramanujan’s notebooks or “lost ” notebook. We also derive a new identity for Eisen-stein series and some curious trigonometric identities. 1. Introduction. Suppose throughout that q = exp(2piiτ), where τ has positive imaginary part, and set (z; q) ∞