Uniform Estimates of Ultraspherical Polynomials of Large Order

http://dx.doi.org/10.4153/CMB-2005-035-2
Canad. Math. Bull. 48(2005), 382-393

  Published:2005-09-01
 Printed: Sep 2005

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 $P_n^{(s)}(x)$.