### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 039, 17 pages      arXiv:1204.0254      https://doi.org/10.3842/SIGMA.2012.039

### Some Remarks on Very-Well-Poised ${}_8\phi_7$ Series

Jasper V. Stokman
Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands

Received April 05, 2012, in final form June 18, 2012; Published online June 27, 2012

Abstract
Nonpolynomial basic hypergeometric eigenfunctions of the Askey-Wilson second order difference operator are known to be expressible as very-well-poised ${}_8\phi_7$ 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 ${}_8\phi_7$ series. We also provide a link to Chalykh's theory on (rank one, BC type) Baker-Akhiezer functions.

Key words: very-well-poised basic hypergeometric series; Askey-Wilson functions; quadratic transformation formulas; theta functions.

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