The Abel Lemma and the q-Gosper Algorithm

Chu has recently shown that the Abel lemma on summations by parts can serve as the underlying relation for Bailey’s 6ψ6 bilateral summation formula. In other words, the Abel lemma spells out the telescoping nature of the 6ψ6 sum. We present a systematic approach to compute Abel pairs for bilateral and unilateral basic hypergeometric summation formulas by using the q-Gosper algorithm. It is demonstrated that Abel pairs can be derived from Gosper pairs. This approach applies to many classical summation formulas.