New bounds for 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 2 nn (λ), with xnn(λ) being the largest zero of the n-th ultraspherical polynomial P (λ) n . For every fixed λ > −1/2, the limit of the ratio of our upper and lower bounds for 1 − x 2 nn (λ) does not exceed 1.6. This paper is a continuation of [1].

Local sign changes of polynomials

“…there is a particularly canonical choice: if we define g to be a bump function suitably localized around the largest root of the Gegenbauer polynomial, this is guaranteed to lead to a function that is compactly supported with support close to 1 and λ m (g) = 0. A result of Driver-Jordaan [12] (see also Nikolov [18]) shows that the largest root of C λ m (x) satisfies…”

“…The bounds in [12,18] are slightly stronger than that (at the level of constants), we have chosen a slightly algebraically easier form for simplicity of exposition. The roots of the Gegenbauer polynomials are simple which means that C (λ) m changes sign in x 1 .…”

Concentration estimates for finite expansions of spherical harmonics on two-point homogeneous spaces via the large sieve principle

“…Precise lower bounds for t n,n have been obtained recently in [17,18]. We are rather interested in an upper bound which can be derived from the Euler-Rayleigh technique.…”

“…We are rather interested in an upper bound which can be derived from the Euler-Rayleigh technique. A simple computation derived from those in [17,18] shows that…”