Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Characterization of Askey-Wilson polynomials   We show that the only monic orthogonal polynomials $\{P_n\}_{n=0}^{\infty}$
that satisfy
$$\pi(x)\mathcal{D}_{q}^2P_{n}(x)=\sum_{j=-2}^{2}a_{n,n+j}P_{n+j}(x),\;
x=\cos\theta,\;~ a_{n,n-2}\neq 0,~ n=2,3,\dots,$$ 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.
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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The categories of this paper are: hep-lat
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The categories of this paper are: q-alg
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-bio
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The categories of this paper are: hep-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: mtrl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-th
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The categories of this paper are: physics
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The categories of this paper are: nlin
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The categories of this paper are: q-fin
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The categories of this paper are: adap-org
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The categories of this paper are: quant-ph