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arXiv · 2111.11663

Asymptotic behaviours of q-orthogonal polynomials from a q-Riemann Hilbert Problem

Nalini Joshi·Tomas Lasic Latimer
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Abstract

We describe a Riemann-Hilbert problem for a family of $q$-orthogonal polynomials, $\{ P_n(x) \}_{n=0}^\infty$, and use it to deduce their asymptotic behaviours in the limit as the degree, $n$, approaches infinity. We find that the $q$-orthogonal polynomials studied in this paper share certain universal behaviours in the limit $n\to\infty$. In particular, we observe that the asymptotic behaviour near the location of their smallest zeros, $x \sim q^{n/2}$, and norm, $\|P_n\|_2$, are independent of the weight function as $n\to\infty$.

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Nalini Joshi, Tomas Lasic Latimer. 2021-11-23. Asymptotic behaviours of q-orthogonal polynomials from a q-Riemann Hilbert Problem. https://doi.org/10.1007/s00365-022-09613-1

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