Irreducibility of the Gegenbauer polynomials

Question

The Gegenbauer polynomials $C_n^{(\alpha)}(x)$ can be defined by requiring that they satisfy that $$ \frac{1}{(1-2xt+t^2)^{\alpha}} = \sum_{n=0}^{\infty} C_n^{(\alpha)}(x)t^n.$$ In the cases when $\alpha \in \mathbb{N}$ is it known whether these polynomials are irreducible over $\mathbb{Q}$? If so does anyone know of either an argument which proves this or a reference? Many Thanks

Answer 1

Well, if you put $x=0$, the lhs is a power series in $t^2$, so $C_n^{(\alpha)}(0)=0$ for odd $n$ and any $\alpha$, so at least for odd $n>1$ the polynomials $C_n^{(\alpha)}(x)$ are reducible as they have a factor of $x$.