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].