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Zbl 1110.30021
Wang, Z.; Wong, R.
Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. (English)
[J] J. Math. Pures Appl. (9) 85, No. 5, 698-718 (2006). ISSN 0021-7824

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.

MSC 2000:: *30E25 Boundary value problems, complex analysis
33C45 Orthogonal polynomials and functions of hypergeometric type
42C05 General theory of orthogonal functions and polynomials

Keywords: Stieltjes-Wigert polynomials; Riemann-Hilbert problem; uniform asymptotics; zeros; logarithmic potential

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