On "mixed" modular equations of degree 21

TL;DR

This paper establishes six new mixed modular equations involving theta functions with specific moduli, contributing to the theory of signature 3 and enabling the evaluation of various theta function quotients.

Contribution

It introduces new modular equations of degree 21 involving theta functions, expanding the understanding of modular identities in signature 3.

Findings

Six new modular equations involving theta functions are established.

Several values of quotients of theta functions are explicitly evaluated.

The work advances the theory of modular identities in signature 3.

Abstract

In the proposed work, we establish a total of six new $P$--$Q$ modular equations involving theta--function $f(-q)$ with moduli of orders 1, 3, 7 and 21.These equations can be regarded as modular identities in the alternate theory of signature 3. As a consequence, several values of quotients of theta--function are evaluated.