A modular type formula for Euler infinite product $(1-x)(1-xq)(1-xq^2)(1-xq^3)...$

TL;DR

This paper introduces a modular type representation for the infinite product (1-x)(1-xq)(1-xq^2)..., unifying various classical modular formulas and providing a new approach to elliptic and modular functions.

Contribution

It presents a novel modular representation for an infinite product, linking it to known modular functions and offering a unified framework for their study.

Findings

Unified modular formula for the infinite product.

Connection to Dedekind's eta, Jacobi theta, and Lambert series.

Provides a new approach to elliptic and modular functions.

Abstract

The main goal of this paper is to give a modular type representation for the infinite product $(1-x)(1-xq)(1-xq^2)(1-xq^3)...$. It is shown that this representation essentially contains the well-known modular formulae either for Dedekind's eta function, Jacobi theta function or for certain Lambert series. Thus a new and unified approach is outlined for the study of elliptic and modular functions and related series.