Some Lambert series expansions of products of theta functions. (English) Zbl 0938.11017

Williams, Kenneth S.

Ramanujan J. 3, No. 4, 367-384 (1999).

Let \(q\) be a complex number satisfying \(|q|=1\) and define the theta function \(\varphi(q)\) by \(\varphi(q)= \sum_{k=-\infty}^\infty q^{x^2}\). Ramanujan has given a number of Lambert series expansions such as \[ \varphi(q) \varphi(q^2)= 1-2 \sum_{n=1}^\infty \frac{(-1)^{\frac{n(n+1)}{2}} q^{2n-1}} {1-q^{2n-1}}. \] A formula is proved which includes this and other expansions as special cases, in fact the theorem proved works for all 101 discriminants \(d\) for which every binary quadratic genus has only one class. The above example is the case \(d=-8\).

Reviewer: M.Peters (Münster)

MSC:

11E16	General binary quadratic forms
33D15	Basic hypergeometric functions in one variable, \({}_r\phi_s\)

Keywords:

theta functions; Lambert series; binary quadratic forms