Asymptotic Expansions of Jacobi Polynomials for Large Values of $\beta$ and of Their Zeros

TL;DR

This paper derives asymptotic expansions for Jacobi polynomials and their zeros when the parameter beta is large, using Laguerre polynomials, and verifies accuracy through numerical examples.

Contribution

It introduces new asymptotic formulas for Jacobi polynomials for large beta, linking them to Laguerre polynomials, with numerical validation.

Findings

Asymptotic expansions accurately approximate Jacobi polynomials for large beta.

Derived formulas relate Jacobi and Laguerre polynomials.

Numerical examples confirm the effectiveness of the approximations.

Abstract

Asymptotic approximations of Jacobi polynomials are given for large values of the $\beta$-parameter and of their zeros. The expansions are given in terms of Laguerre polynomials and of their zeros. The levels of accuracy of the approximations are verified by numerical examples.