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Uniform Estimates of Ultraspherical Polynomials of Large Order
Published online by Cambridge University Press: 20 November 2018
Abstract
In this paper we prove the sharp inequality
$$\left| P_{n}^{\left( s \right)}\left( x \right) \right|\le P_{n}^{\left( s \right)}\left( 1 \right)\left( {{\left| x \right|}^{n}}+\frac{n-1}{2s+1}\left( 1-{{\left| x \right|}^{n}} \right) \right)$$
where $P_{n}^{\left( s \right)}\left( x \right)$ is the classical ultraspherical polynomial of degree
$n$ and order
$s\ge n\frac{1+\sqrt{5}}{4}$. This inequality can be refined in
$\left[ 0,z_{n}^{s} \right]$ and
$\left[ z_{n}^{s},1 \right]$, where
$z_{n}^{s}$ denotes the largest zero of
$P_{n}^{\left( s \right)}\left( x \right)$.
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- Research Article
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- Copyright
- Copyright © Canadian Mathematical Society 2005
References
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