# A New Summation Formula for WP-Bailey Pairs

## TL;DR

This paper introduces a new summation formula for WP-Bailey pairs, deriving novel hypergeometric identities and theta series formulas through limiting cases and transformations, expanding the theoretical framework of basic hypergeometric series.

## Contribution

It presents a new summation formula for WP-Bailey pairs and derives related hypergeometric and theta series identities using limiting cases of known transformations.

## Key findings

- New basic hypergeometric summation and transformation formulas.
- Derived identities for theta series and Lambert series.
- Extension of WP-Bailey pair theory with new summation techniques.

## Abstract

Let $(\alpha_n(a,k),\beta_n(a,k))$ be a WP-Bailey pair. Assuming the limits exist, let \[ (\alpha_n^*(a),\beta_n^*(a))_{n\geq 1} = \lim_{k \to 1}\left(\alpha_n(a,k),\frac{\beta_n(a,k)}{1-k}\right)_{n\geq 1} \] be the \emph{derived} WP-Bailey pair. By considering a particular limiting case of a transformation due to George Andrews, we derive new basic hypergeometric summation and transformation formulae involving derived WP-Bailey pairs. We then use these formulae to derive new identities for various theta series/products which are expressible in terms of certain types of Lambert series.