location:  Publications → journals → CMB
Abstract view

# Uniform Estimates of Ultraspherical Polynomials of Large Order

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