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Uniform Estimates of Ultraspherical Polynomials of Large Order

Published online by Cambridge University Press:  20 November 2018

Laura De Carli
Laura De Carli*
Affiliation:
Department of Mathematics, Florida International University, University Park, Miami, FL 33199, U.S.A. e-mail: decarlil@fiu.edu
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Abstract

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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 $P_{n}^{\left( s \right)}\left( x \right)$.

Keywords

42C0533C47
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 3 , 01 September 2005 , pp. 382 - 393
Copyright
Copyright © Canadian Mathematical Society 2005

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