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Askey-Wilson polynomials and a double $ q$-series transformation formula with twelve parameters


Author: Zhi-Guo Liu
Journal: Proc. Amer. Math. Soc. 147 (2019), 2349-2363
MSC (2010): Primary 05A30, 33D15, 33D45, 11E25
DOI: https://doi.org/10.1090/proc/14411
Published electronically: February 20, 2019
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Abstract: The Askey-Wilson polynomials are the most general classical orthogonal polynomials that are known, and the Nassrallah-Rahman integral is a very general extension of Euler's integral representation of the classical $ _2F_1$ function. Based on a $ q$-series transformation formula and the Nassrallah-Rahman integral we prove a $ q$-beta integral which has twelve parameters, with several other results, both classical and new, included as special cases. This $ q$-beta integral also allows us to derive a curious double $ q$-series transformation formula, which includes one formula of Al-Salam and Ismail as a special case.


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Additional Information

Zhi-Guo Liu
Affiliation: School of Mathematical Sciences and S Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Road, Shanghai 200241, People’s Republic of China
Email: zgliu@math.ecnu.edu.cn; liuzg@hotmail.com

DOI: https://doi.org/10.1090/proc/14411
Keywords: Askey--Wilson integral, Askey--Wilson polynomials, Nassrallah--Rahman integral, $q$-beta integral
Received by editor(s): June 21, 2018
Received by editor(s) in revised form: September 8, 2018, September 9, 2018, September 25, 2018, and September 26, 2018
Published electronically: February 20, 2019
Additional Notes: The author was supported in part by the National Natural Science Foundation of China and Science and Technology Commission of Shanghai Municipality (Grant No. 13dz2260400)
Communicated by: Mourad Ismail
Article copyright: © Copyright 2019 American Mathematical Society