On the extreme zeros of Jacobi polynomials

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 $1-x_{nn}^2(\lambda )$, with $x_{nn}(\lambda )$ being the largest zero of the $n$-th ultraspherical polynomial $P_n^{(\lambda )}$. For every fixed $\lambda >-1/2$, the limit of the ratio of our upper and lower bounds for $1-x_{nn}^2(\lambda )$ does not exceed $1.6$. This paper is a continuation of [1].

12 pages, 1 figure