On some P-Q mixed modular equations of degree 5

TL;DR

This paper derives new P-Q modular equations of degree 5 related to Ramanujan's theta-function identities, enabling the evaluation of specific theta-function quotients.

Contribution

It introduces novel modular equations analogous to Ramanujan's, expanding the understanding of theta-function relations of degree 5.

Findings

New P-Q modular equations of degree 5 derived

Values of certain theta-function quotients explicitly evaluated

Enhanced understanding of Ramanujan's theta-function identities

Abstract

In his second notebook, Ramanujan recorded total of 23 P-Q modular equations involving theta-functions $f(-q)$, $\varphi(q)$ and $\psi(q)$. In this paper, modular equations analogous to those recorded by Ramanujan are obtained involving $f(-q)$. As a consequence, values of certain quotients of theta-function are evaluated.