The t-coefficient method to partial theta function identities and Ramanujan's 1ψ1 summation formula

This paper is concerned with a uniform approach-the t-coefficient method-to basic hypergeometric series. By virtue of this method, some new and elementary proofs for four partial theta function identities due to Alladi and Berkovich (2004) [8], Andrews and Warnaar (2007) [4], Berkovich [5] and Warnaar (2003) [2], as well as for Ramanujan's 1ψ1 summation formula are obtained.