On the extreme zeros of Jacobi polynomials

2020/02/07 by Geno Nikolov, Nikolov, Geno
Mathematics · Physics and Astronomy · #33C45 #42C05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Numerical methods in inverse problems #Quantum Mechanics and Non-Hermitian Physics

paper · pdf · doi:10.48550/arxiv.2002.02633

openalex publication_date 2020/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

By applying the Euler--Rayleigh methods to a specific representation of the Jacobi polynomials as hypergeometric functions, we obtain new bounds for their largest zeros. In particular, we derive upper and lower bound for 1-x_nn²(λ), with x_nn(λ) being the largest zero of the n-th ultraspherical polynomial P_n^(λ). For every fixed λ>-1/2, the limit of the ratio of our upper and lower bounds for 1-x_nn²(λ) does not exceed 1.6. This paper is a continuation of [1].

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