In this article, we use identities found in N.J. Fine’s book Basic hypergeometric series and its applications to derive a one-parameter generalization of the product of two partial theta functions discovered by G.E. Andrews and S.O. Warnaar. We also give two different proofs of this generalization, one of which is motivated by the work of A. Berkovich and the other is given by M.E.H. Ismail.