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A characterization of Askey-Wilson polynomials

Authors: Maurice Kenfack Nangho and Kerstin Jordaan
Journal: Proc. Amer. Math. Soc. 147 (2019), 2465-2480
MSC (2010): Primary 33D45; Secondary 33C45
Published electronically: March 1, 2019
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Abstract: We show that the only monic orthogonal polynomials $ \{P_n\}_{n=0}^{\infty }$ that satisfy

$\displaystyle \pi (x)\mathcal {D}_{q}^2P_{n}(x)=\sum _{j=-2}^{2}a_{n,n+j}P_{n+j}(x),\; x=\cos \theta ,\;~ a_{n,n-2}\neq 0,~ n=2,3,\dots ,$    

where $ \pi (x)$ is a polynomial of degree at most $ 4$ and $ \mathcal {D}_{q}$ is the Askey-Wilson operator, are Askey-Wilson polynomials and their special or limiting cases as one or more parameters tends to $ \infty $. This completes and proves a conjecture by Ismail concerning a structure relation satisfied by Askey-Wilson polynomials. We use the structure relation to derive upper bounds for the smallest zero and lower bounds for the largest zero of Askey-Wilson polynomials and their special cases.

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Additional Information

Maurice Kenfack Nangho
Affiliation: Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, 0003, South Africa
Address at time of publication: Department of Mathematics and Computer Science, University of Dschang, Dschang, Cameroon

Kerstin Jordaan
Affiliation: Department of Decision Sciences, University of South Africa, Pretoria, 0003, South Africa

Received by editor(s): December 4, 2017
Received by editor(s) in revised form: June 28, 2018
Published electronically: March 1, 2019
Additional Notes: The research of the first author was supported by a Vice-Chancellor’s Postdoctoral Fellowship from the University of Pretoria.
The second author served as corresponding author for this paper. The research by the second author was partially supported by the National Research Foundation of South Africa under grant number 108763.
Communicated by: Mourad Ismail
Article copyright: © Copyright 2019 American Mathematical Society