A characterization of Askey-Wilson polynomials

Kenfack Nangho, Maurice; Jordaan, Kerstin

doi:10.1090/proc/14317

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

Abstract:

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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