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# Uniform Estimates of Ultraspherical Polynomials of Large Order

Published:2005-09-01
Printed: Sep 2005
• Laura De Carli
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## Abstract

In this paper we prove the sharp inequality $$|P_n^{(s)}(x)|\leq P_n^{(s)}(1)\bigl(|x|^n +\frac{n-1}{2 s+1}(1-|x|^n)\bigr),$$ where $P_n^{(s)}(x)$ is the classical ultraspherical polynomial of degree $n$ and order $s\ge n\frac{1+\sqrt 5}{4}$. This inequality can be refined in $[0,z_n^s]$ and $[z_n^s,1]$, where $z_n^s$ denotes the largest zero of $P_n^{(s)}(x)$.
 MSC Classifications: 42C05 - Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45] 33C47 - Other special orthogonal polynomials and functions