Roots of a Gegenbauer polynomial
= Roots of a Gegenbauer polynomial
{c}
For $\alpha>0$, all $n$ roots of $C_n^\alpha$ are simple and lie in $(-1,1)$. If the product $q$ of its interior linear factors had degree below $n$, orthogonality would make $\int C_n^\alpha q w=0$, while $C_n^\alpha q=q^2(C_n^\alpha/q)$ has a constant strict sign away from finitely many points.