# A Reciprocity Relation for WP-Bailey Pairs

## TL;DR

This paper introduces a new transformation for WP-Bailey pairs, leading to novel summation formulas and extending classical identities by Jacobi and Ramanujan through the analysis of WP-Bailey chains.

## Contribution

It presents a new general transformation for WP-Bailey pairs derived from a limiting case of a WP-Bailey chain, expanding the toolkit for identities in q-series.

## Key findings

- New summation formulas involving WP-Bailey pairs
- New proofs of classical identities by Jacobi and Ramanujan
- Extensions of classical identities to specializations of WP-Bailey pairs

## Abstract

We derive a new general transformation for WP-Bailey pairs by considering the a certain limiting case of a WP-Bailey chain previously found by the authors, and examine several consequences of this new transformation. These consequences include new summation formulae involving WP-Bailey pairs. Other consequences include new proofs of some classical identities due to Jacobi, Ramanujan and others, and indeed extend these identities to identities involving particular specializations of arbitrary WP-Bailey pairs.