Some Lambert Series Expansions of Products of Theta Functions

Let q be a complex number satisfying |q| < 1. The theta function φ(q) is defined by φ(q) = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\sum\nolimits_{x = - \infty }^\infty {q^{x^2 } }$$ \end{document}. Ramanujan has given a number of Lambert series expansions such as\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\phi (q)\phi (q^2 ) = 1 - 2\sum\limits_{n = 1}^\infty {\frac{{( - 1)^{n(n + 1)/2} q^{2n - 1} }}{{1 - q^{2n - 1} }}}$$ \end{document} A formula is proved which includes this and other expansions as special cases.