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A quadratic formula for basic hypergeometric series related to Askey-Wilson polynomials


Authors: Victor J. W. Guo, Masao Ishikawa, Hiroyuki Tagawa and Jiang Zeng
Journal: Proc. Amer. Math. Soc. 143 (2015), 2003-2015
MSC (2010): Primary 33C45, 33D45; Secondary 05A19
DOI: https://doi.org/10.1090/S0002-9939-2015-12099-0
Published electronically: January 22, 2015
MathSciNet review: 3314110
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Abstract: We prove a general quadratic formula for basic hypergeometric series, from which simple proofs of several recent determinant and Pfaffian formulas are obtained. A special case of the quadratic formula is actually related to a Gram determinant formula for Askey-Wilson polynomials. We also show how to derive a recent double-sum formula for the moments of Askey-Wilson polynomials from Newton's interpolation formula.


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Additional Information

Victor J. W. Guo
Affiliation: Department of Mathematics, East China Normal University, Shanghai 200062, People’s Republic of China
Email: jwguo@math.ecnu.edu.cn

Masao Ishikawa
Affiliation: Department of Mathematics, Faculty of Education, University of the Ryukyus, Nishihara, Okinawa 901-0213, Japan
Email: ishikawa@edu.u-ryukyu.ac.jp

Hiroyuki Tagawa
Affiliation: Department of Mathematics, Faculty of Education, Wakayama University, Sakaedani, Wakayama 640-8510, Japan
Email: tagawa@math.edu.wakayama-u.ac.jp

Jiang Zeng
Affiliation: Université de Lyon, Université Lyon 1, Institut Camille Jordan, UMR 5208 du CNRS, 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France
Email: zeng@math.univ-lyon1.fr

DOI: https://doi.org/10.1090/S0002-9939-2015-12099-0
Keywords: Quadratic formula of basic hypergeometric series, Askey-Wilson polynomials, moments of Askey-Wilson polynomials, Gram determinants, Pfaffians, Desnanot--Jacobi adjoint matrix theorem.
Received by editor(s): November 7, 2012
Received by editor(s) in revised form: July 17, 2013
Published electronically: January 22, 2015
Additional Notes: This work was partially supported by JSPS Grant-in-Aid for Scientific Research (C) 23540017.
The fourth author was partially supported by CMIRA COOPERA 2012 de la Région Rhône-Alpes.
Dedicated: Dedicated to Srinivasa Ramanujan on the occasion of his 125th birth anniversary
Communicated by: Sergei K. Suslov
Article copyright: © Copyright 2015 American Mathematical Society