Lifting Bailey Pairs to WP-Bailey Pairs (1901.04841v1)

Abstract: A pair of sequences $(\alpha_{n}(a,k,q),\beta_{n}(a,k,q))$ such that $\alpha_0(a,k,q)=1$ and [ \beta_{n}(a,k,q) = \sum_{j=0}^{n} \frac{(k/a; q){n-j}(k; q){n+j}}{(q;q){n-j}(aq;q){n+j}}\alpha_{j}(a,k,q) ] is termed a \emph{WP-Bailey Pair}. Upon setting $k=0$ in such a pair we obtain a \emph{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.