Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials
SIAM Journal on Mathematical Analysis - Special issue: the articles in this issue are dedicated to Richard Askey and Frank Olver
Reflection coefficients for the generalized Jacobi weight functions
Journal of Approximation Theory
A Riemann-Hilbert approach to asymptotic questions for orthogonal polynomials
Journal of Computational and Applied Mathematics - Special issue on orthogonal polynomials, special functions and their applications
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Global asymptotics of orthogonal polynomials associated with |x|2αe-Q(x)
Journal of Approximation Theory
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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 (RH) problem for orthogonal polynomials. Using the steepest descent method of Deift and Zhou, we analyze the RH 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 RH problem will be the local analysis around the algebraic singularities, for which we use Bessel functions of appropriate order. In addition, the RH approach gives us also strong asymptotics of the orthogonal polynomials near the algebraic singularities in terms of Bessel functions.