Evaluations of certain theta functions in Ramanujan theory of alternative modular bases

TL;DR

This paper evaluates specific Borwein theta functions within Ramanujan's alternative modular bases, extending existing theory and proposing new evaluation theorems and conjectures for a broad class of these functions.

Contribution

It provides new evaluations and conjectures for Borwein's theta functions in the context of Ramanujan's alternative elliptic modular bases, expanding the theoretical framework.

Findings

Evaluations of certain Borwein theta functions

Development of new evaluation theorems

Proposals of conjectures for general classes

Abstract

We give evaluations of certain Borwein's theta functions which appear in Ramanujan theory of alternative elliptic modular bases. Most of this theory where developed by B.C. Berndt, S. Bhargava and F.G. Garvan. We also study the most general class of these theta functions and give evaluation theorems and conjectures.