A modular type formula for Euler infinite product $(1-x)(1-xq)(1-x{q}^2)(1-x{q}^3)\mathrm{.}\mathrm{.}\mathrm{.}$

The main goal of this paper is to give a modular type representation for the infinite product $(1-x)(1-xq)(1-x{q}^2)(1-x{q}^3)\mathrm{.}\mathrm{.}\mathrm{.}$. 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.