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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A characterization of Askey-Wilson polynomials
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by Maurice Kenfack Nangho and Kerstin Jordaan PDF
Proc. Amer. Math. Soc. 147 (2019), 2465-2480 Request permission


We show that the only monic orthogonal polynomials $\{P_n\}_{n=0}^{\infty }$ that satisfy \begin{equation*} \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 , \end{equation*} 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
  • Email:,
  • Kerstin Jordaan
  • Affiliation: Department of Decision Sciences, University of South Africa, Pretoria, 0003, South Africa
  • MR Author ID: 701645
  • Email:
  • 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
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 2465-2480
  • MSC (2010): Primary 33D45; Secondary 33C45
  • DOI:
  • MathSciNet review: 3951425