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Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. (English) Zbl 1110.30021
Summary: It has been known for some time that the existing asymptotic methods for integrals and differential equations are not applicable in the case of Stieltjes-Wigert polynomials with degree going to infinity. Using the recently introduced nonlinear steepest descent method for Riemann-Hilbert problems, here we not only derive an asymptotic expansion for these polynomials, but we also show that the result holds uniformly in the complex plane except for a sector containing the real axis from $-\infty$ to $\frac {1}{4}$. Furthermore, we give an asymptotic formula for the zeros of these polynomials, which approximates the true values of the zeros closely.

30E25Boundary value problems, complex analysis
33C45Orthogonal polynomials and functions of hypergeometric type
42C05General theory of orthogonal functions and polynomials
Full Text: DOI
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