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arXiv · 2311.12761

Lecture hall graphs and the Askey scheme

Sylvie Corteel·Bhargavi Jonnadula·Jonathan P. Keating·Jang Soo Kim
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Abstract

We establish, for every family of orthogonal polynomials in the $ q $-Askey scheme and the Askey scheme, a combinatorial model for mixed moments and coefficients in terms of paths on the lecture hall graph. This generalizes the previous results of Corteel and Kim for the little $ q $-Jacobi polynomials. We build these combinatorial models by bootstrapping, beginning with polynomials at the bottom and working towards Askey--Wilson polynomials which sit at the top of the $ q $-Askey scheme. As an application of the theory, we provide the first combinatorial proof of the symmetries in the parameters of the Askey--Wilson polynomials.

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Sylvie Corteel, Bhargavi Jonnadula, Jonathan P. Keating, Jang Soo Kim. 2023-11-21. Lecture hall graphs and the Askey scheme. https://doi.org/10.1016/j.aim.2026.111056

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