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On the extreme zeros of Jacobi polynomials

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arxiv 2002.02633 v1 pith:3HQIMZQS submitted 2020-02-07 math.CA

classification math.CA
keywords lambdaboundsjacobilargestlowerpolynomialsupperzeros
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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].

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  1. One Discrete Gaussian Sample in $2^{n/2+o(n)}$ Time

    cs.DS 2026-08 conditional novelty 8.0 of 10

    One discrete Gaussian sample at an arbitrary parameter can be drawn in 2^(n/2+o(n)) expected time, resolving an open question from ADRS15.

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