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.

We describe a Riemann-Hilbert problem for a family of q-orthogonal polynomials, {Pn(x)} ∞ n=0 , 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 → ∞. In particular, we observe that the asymptotic behaviour near the location of their smallest zeros, x ∼ q n/2 , and norm, Pn 2 , are independent of the weight function as n → ∞.

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