# General WP-Bailey Chains

## TL;DR

This paper introduces a unified framework for WP-Bailey chains, showing how many existing chains are special cases and discovering new branches and transformation formulas for basic hypergeometric series.

## Contribution

It presents a general theory of WP-Bailey chains, unifies previous results, and introduces new branches and transformation formulas for hypergeometric series.

## Key findings

- Many existing WP-Bailey chains are special cases of the new framework.
- Three new branches of the WP-Bailey tree are identified.
- New transformation formulas for basic hypergeometric series are derived.

## Abstract

Motivated by a recent paper of Liu and Ma, we describe a number of general WP-Bailey chains. We show that many of the existing WP-Bailey chains (or branches of the WP-Bailey tree), including chains found by Andrews, Warnaar and Liu and Ma, arise as special cases of these general WP-Bailey chains. We exhibit three new branches of the WP-Bailey tree, branches which also follow as special cases of these general WP-Bailey chains. Finally, we describe a number of new transformation formulae for basic hypergeometric series which arise as consequences of these new WP-Bailey chains.