The -coefficient method to partial theta function identities and Ramanujan’s 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 1 ψ 1 summation formula are obtained.