Asymptotic expansions of Jacobi polynomials and of the nodes and weights of Gauss-Jacobi quadrature for large degree and parameters in terms of elementary functions

2020/07/21 by Amparo Gil, Javier Segura, Gil, Amparo +3 · 1 citation
Mathematics · #33C45 #41A60 #65D32 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Fractional Differential Equations Solutions #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2007.10748

openalex publication_date 2020/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Asymptotic approximations of Jacobi polynomials are given in terms of elementary functions for large degree n and parameters α and β. From these new results, asymptotic expansions of the zeros are derived and methods are given to obtain the coefficients in the expansions. These approximations can be used as initial values in iterative methods for computing the nodes of Gauss--Jacobi quadrature for large degree and parameters. The performance of the asymptotic approximations for computing the nodes and weights of these Gaussian quadratures is illustrated with numerical examples.

Citations

• Fast, reliable and unrestricted iterative computation of Gauss–Hermite and Gauss–Laguerre quadratures

Cited by

• Fast and accurate computation of classical Gaussian quadratures

Related

• Non-iterative computation of Gauss-Jacobi quadrature
• Asymptotic computation of classical orthogonal polynomials
• Uniform asymptotic expansions of the Jacobi polynomials and an associated function
• Asymptotic approximations to the nodes and weights of Gauss-Hermite and Gauss-Laguerre quadratures
• Error bounds for the asymptotic expansions of the Jacobi polynomials