2019
DOI: 10.1090/proc/14370
|View full text |Cite
|
Sign up to set email alerts
|

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].

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
2
2

Citation Types

0
4
0

Year Published

2020
2020
2024
2024

Publication Types

Select...
4
2
1

Relationship

0
7

Authors

Journals

citations
Cited by 8 publications
(4 citation statements)
references
References 19 publications
0
4
0
Order By: Relevance
“…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…”
Section: Proof Of Theoremmentioning
confidence: 99%
See 1 more Smart Citation
“…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…”
Section: Proof Of Theoremmentioning
confidence: 99%
“…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 .…”
Section: Proof Of Theoremmentioning
confidence: 99%
“…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.…”
Section: Preliminariesmentioning
confidence: 99%
“…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…”
Section: Preliminariesmentioning
confidence: 99%