  In this paper, by applying a range of classic summation and transformation
formulas for basic hypergeometric series, we obtain a three-term identity for
partial theta functions. It extends the Andrews-Warnaar partial theta function
identity, and also unifies several results on partial theta functions due to
Ramanujan, Lovejoy and Kim. We also establish a two-term version of the
extension, which can be used to derive identities for partial and false theta
functions. Finally, we present a relation between the big $q$-Jacobi
polynomials and the Andrews-Warnaar partial theta function identity.
