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

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, $\Vert P_n{\Vert }_2$, are independent of the weight function as $n\to \infty$.

23 pages, 1 figure

References in corpus (3)

• Discrete quantum model of the harmonic oscillator
• Quantum state transfer in spin chains with q-deformed interaction terms
• On a class of q-orthogonal polynomials and the q-Riemann Hilbert Problem

Cited by in corpus (1)

• Asymptotics for multiple q-orthogonal polynomials from the RHP