Asymptotics for the simplest generalized Jacobi polynomials recurrence coefficients from Freud's equations: numerical explorations.

TL;DR

This paper investigates how algebraic singularities in the weight function affect the asymptotic behavior of recurrence coefficients in generalized Jacobi polynomials, combining theoretical analysis with numerical exploration.

Contribution

It provides new insights into the asymptotic behavior of recurrence coefficients for generalized Jacobi polynomials with singularities, supported by numerical experiments.

Findings

Singular points significantly influence recurrence coefficient asymptotics

Numerical methods confirm theoretical asymptotic predictions

Enhanced understanding of polynomial behavior near singularities

Abstract

Generalized Jacobi polynomials are orthogonal polynomials related to a weight function which is smooth and positive on the whole interval of orthogonality up to a finite number of points, where algebraic singularities occur. The influence of these singular points on the asymptotic behaviour of the recurrence coefficients is investigated.