Gegenbauer polynomial ($C_n^{\alpha }(x)$)

For $\alpha >0$, the Gegenbauer polynomial $C_n^{\alpha }$ is the degree-$n$ polynomial solution ofThese polynomials are orthogonal on $(-1,1)$ with weight $(1-x^2{)}^{\alpha -1/2}$.

The Gegenbauer differential equation is the displayed Sturm–Liouville eigenvalue equation defining the Gegenbauer polynomial. Multiplication by $(1-x^2{)}^{\alpha -1/2}$ puts it in self-adjoint form.

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 $\int C_n^{\alpha }qw=0$, while $C_n^{\alpha }q=q^2(C_n^{\alpha }/q)$ has a constant strict sign away from finitely many points.