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 SIGMA, 2012, Volume 8, 039, 17 pages (Mi sigma716)

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

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.

Keywords: very-well-poised basic hypergeometric series, Askey–Wilson functions, quadratic transformation formulas, theta functions.

DOI: https://doi.org/10.3842/SIGMA.2012.039

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ArXiv: 1204.0254
MSC: 33D15; 33D45
Received: April 5, 2012; in final form June 18, 2012; Published online June 27, 2012
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Citation: Jasper V. Stokman, “Some remarks on very-well-poised $_8\phi_7$ series”, SIGMA, 8 (2012), 039, 17 pp.

Citation in format AMSBIB
\Bibitem{Sto12} \by Jasper V. Stokman \paper Some remarks on very-well-poised ${}_8\phi_7$ series \jour SIGMA \yr 2012 \vol 8 \papernumber 039 \totalpages 17 \mathnet{http://mi.mathnet.ru/sigma716} \crossref{https://doi.org/10.3842/SIGMA.2012.039} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2946861} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000305677000001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84864532874} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Chalykh O., Etingof P., “Orthogonality Relations and Cherednik Identities for Multivariable Baker-Akhiezer Functions”, Adv. Math., 238 (2013), 246–289
2. Stokman J.V., “Connection Coefficients for Basic Harish-Chandra Series”, Adv. Math., 250 (2014), 351–386
3. Stokman J.V., “The C-Function Expansion of a Basic Hypergeometric Function Associated to Root Systems”, Ann. Math., 179:1 (2014), 253–299
4. Stokman J.V., “Connection Problems For Quantum Affine Kz Equations and Integrable Lattice Models”, Commun. Math. Phys., 338:3 (2015), 1363–1409
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