Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight

M. Vanlessen

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arXiv · math/0212014

Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight

M. Vanlessen

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Abstract

We study asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight, which is a weight function with a finite number of algebraic singularities on $[-1,1]$. The recurrence coefficients can be written in terms of the solution of the corresponding Riemann-Hilbert problem for orthogonal polynomials. Using the steepest descent method of Deift and Zhou, we analyze the Riemann-Hilbert problem, and obtain complete asymptotic expansions of the recurrence coefficients. We will determine explicitly the order $1/n$ terms in the expansions. A critical step in the analysis of the Riemann-Hilbert problem will be the local analysis around the algebraic singularities, for which we use Bessel functions of appropriate order.

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BibTeX RIS

M. Vanlessen. 2002-12-02. Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight. https://arxiv.org/abs/math/0212014

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