On a new modular equation of degree five and Eisenstein series identities of associated levels

TL;DR

This paper derives new theta function identities at level ten, introduces a novel modular equation of degree five, and establishes Eisenstein series identities of level ten using advanced summation formulas.

Contribution

It presents a new modular equation of degree five and Eisenstein series identities of level ten based on theta function identities and summation formulas.

Findings

Derived theta function identities of level ten using Ramanujan's summation formula

Established a new modular equation of degree five

Proved Eisenstein series identities of level ten using Bailey's summation formula

Abstract

In this research article, we obtain few theta function identities of level ten employing Ramanujan's $_1 \psi_1$ summation formula. Using these identities, we derive a new modular equation of degree five. Further, we establish Eisenstein series identities of level ten using Bailey's very-well poised $_6 \psi_6$ summation formula.