On a class of q -orthogonal polynomials and the q -Riemann–Hilbert problem

Abstract

We give an explicit solution of a q -Riemann–Hilbert problem that arises in the theory of orthogonal polynomials, prove that it is unique and deduce several properties. In particular, we describe the asymptotic behaviour of zeros in the limit as the degree of the polynomial approaches infinity.

Asymptotics of discrete q-Freud II orthogonal polynomials from the q-Riemann Hilbert problem

“…Remark 2.4. It can be shown that solution of the RHP given by definition 2.1 satisfies a Lax pair, namely a q-difference equation in z and a recurrence relation for the parameter n. The proof follows using the same arguments as in [5].…”

“…In this paper, we described the asymptotic behaviour of a class of qF II polynomials, defined in section 1.3, by using the q-RHP setting [5]. Our main results are theorems 1.5, 1.7 and 1.8.…”

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

“…In this paper, we determined the asymptotic behaviour of a general class of qorthogonal polynomials by using the q-RHP setting [19]. The work is motivated by the methods developed by Deift et al [9], which used the RHP setting to determine the asymptotic behaviour of semi-classical orthogonal polynomials.…”

“…Extending on our earlier theory [19], in this paper we will use a RHP to obtain detailed asymptotic results for a large class of q-orthogonal polynomials. We will observe an interesting intersection with q-RHP theory and provide explicit examples highlighting aspects of q-RHP theory discussed in the literature [26,29,1].…”

“…Note that h 0 (z) is equivalent to the definition of h(z) given in [19,Definition 1.3]. Consequently, we define the q-RHP: Definition 1.4 (q-RHP).…”