# Lifting Bailey Pairs to WP-Bailey Pairs

## TL;DR

This paper explores how to extend Bailey pairs to WP-Bailey pairs, leading to new identities in basic hypergeometric series and Rogers-Ramanujan-Slater type sums.

## Contribution

It introduces a method for lifting Bailey pairs to WP-Bailey pairs and derives new identities from these extended pairs.

## Key findings

- New WP-Bailey pairs constructed
- New identities between hypergeometric series derived
- Single and double sum identities of Rogers-Ramanujan-Slater type obtained

## 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.