An expansion formula for q-series and applications

In this paper the author proves a q-expansion formula which utilizes the Leibniz formula for the q-differential operator. This expansion leads to new proofs of the Rogers - Fine identity, the nonterminating ₆φ ₅ summation formula, and Watson's q-analog of Whipple's theorem. Andrews' identities for sums of three squares and sums of three triangular numbers are also derived. Other identities of Andrews and new identities for Hecke type series are also discussed.