Geodesic
Uniform Estimates of Ultraspherical Polynomials of Large Order
Canadian mathematical bulletin, Tome 48 (2005) no. 3, pp. 382-393

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DOI

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 $P_{n}^{\left( s \right)}\left( x \right)$ .
DOI : 10.4153/CMB-2005-035-2
Mots-clés : 42C05, 33C47 
Carli, Laura De. Uniform Estimates of Ultraspherical Polynomials of Large Order. Canadian mathematical bulletin, Tome 48 (2005) no. 3, pp. 382-393. doi: 10.4153/CMB-2005-035-2
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     author = {Carli, Laura De},
     title = {Uniform {Estimates} of {Ultraspherical} {Polynomials} of {Large} {Order}},
     journal = {Canadian mathematical bulletin},
     pages = {382--393},
     year = {2005},
     volume = {48},
     number = {3},
     doi = {10.4153/CMB-2005-035-2},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2005-035-2/}
}
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