Lifting Bailey pairs to WP-Bailey pairs

Abstract

A pair of sequences ( α n ( a , k , q ) , β n ( a , k , q ) ) such that α 0 ( a , k , q ) = 1 and β n ( a , k , q ) = ∑ j = 0 n ( k / a ; q ) n − j ( k ; q ) n + j ( q ; q ) n − j ( a q ; q ) n + j α j ( a , k , q ) is termed a WP-Bailey Pair. Upon setting k = 0 in such a pair we obtain a Bailey pair. In the present paper we consider the problem of “lifting” a Bailey pair to a WP-Bailey pair, and use some of the new WP-Bailey pairs found in this way to derive some new identities between basic hypergeometric series and new single-sum and double-sum identities of the Rogers–Ramanujan–Slater type.