On Eisenstein series identities and new identities connecting Ramanujan-G\"ollnitz-Gordon continued fraction and Ramanujan's continued fraction of order four

TL;DR

This paper derives new identities linking Ramanujan-G"ollnitz-Gordon and Ramanujan's continued fractions using theta functions and Eisenstein series, expanding the understanding of q-series and special functions.

Contribution

It introduces novel identities connecting specific continued fractions and Eisenstein series through classical q-series and theta function techniques.

Findings

Identities connecting Ramanujan-G"ollnitz-Gordon and Ramanujan's continued fractions.

Eisenstein series identities derived from Ramanujan's $_1 \\psi_1$ summation.

Application of Jacobi's theta function product expansion to establish new relations.

Abstract

By employing the classical tools from the theory of $q$-series and theta functions, new fascinating identities on different continued fractions can be achieved. In this article, we use the product expansion of Jacobi's theta function to establish identities that connect Ramanujan-G\"ollnitz-Gordon continued fraction with Ramanujan's continued fraction of order four. Also, we obtain Eisenstein series identities using Ramanujan's $_1 \psi_1$ summation formula.