Some Remarks on Very-Well-Poised 8phi7 Series

TL;DR

This paper explores the properties of very-well-poised 8phi7 basic hypergeometric series, deriving new identities and connecting them to existing literature and Baker-Akhiezer functions.

Contribution

It introduces new identities for 8phi7 series and links hypergeometric functions to Baker-Akhiezer functions, expanding theoretical understanding.

Findings

Derived new hypergeometric identities

Connected 8phi7 series to Baker-Akhiezer functions

Provided a new derivation of a quadratic transformation

Abstract

Nonpolynomial basic hypergeometric eigenfunctions of the Askey-Wilson second order difference operator are known to be expressible as very-well-poised 8phi7 series. In this paper we use this fact to derive various basic hypergeometric and theta function identities. We relate most of them to identities from the existing literature on basic hypergeometric series. This leads for example to a new derivation of a known quadratic transformation formula for very-well-poised 8phi7 series. We also provide a link to Chalykh's theory on (rank one, BC type) Baker-Akhiezer functions.