The product of partial theta functions (Q2383533)

With the standard notation for basic hypergeometric series, the authors prove in this paper the following identity (1) \[ \Big(\sum_{n=0}^{\infty}(-1)^n\,a^{n}\,q^{n(n-1)/2}\Big)\,\Big(\sum_{n=0}^{\infty}(-1)^n\,b^{n}\,q^{n(n-1)/2}\Big) =(q)_{\infty}(a)_{\infty}(b)_{\infty}\,\sum_{n=0}^{\infty}\frac{(abq^{n-1})_{n}\,q^{n}}{(q)_{n}(a)_{n}(b)_{n}}\,. \] Series of the form appearing in the left hand side of (1) are called partial theta functions. The authors deduce from (1) a generalized triple product identity proved by the second named author in [''Partial theta functions. I: Beyond the lost notebook.'' Proc. Lond. Math. Soc., 87, No.2, 363--395 (2003; Zbl 1089.05009)]. They also discuss the relationship of (1) to other theorems in \(q\)-hypergeometric series.