A New Summation Formula for WP-Bailey Pairs

Abstract

Let $(α_n(a,k),β_n(a,k))$ be a WP-Bailey pair. Assuming the limits exist, let \[ (α_n^*(a),β_n^*(a))_{n\geq 1} = \lim_{k \to 1}\left(α_n(a,k),\frac{β_n(a,k)}{1-k}\right)_{n\geq 1} \] 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.