Looking for hardcore orthogonal polynomial people here...
If we hold the degree $\ell$ constant and take the order $\alpha$ to infinity, the Gegenbauer polynomial $G_\ell^{(\alpha)}$ approaches the Hermite polynomial of degree $\ell$.
At another extreme, if we hold $\alpha$ constant and take $\ell$ to infinity, we approach the Chebyshev polynomial; see e.g. 8.21.18 in Szegö's book on orthogonal polynomials.
I'm working in a regime where $\alpha$ is large, and $\ell$ is large but much smaller than $\alpha$: say $\ell = O(\sqrt{\alpha})$ or even $\ell=O(\log \alpha)$. In this case the Hermite approximation seems to break down. Or to put it better: how quickly does the Gegenbauer polynomial approach the Hermite polynomial, especially in the interior of the interval, say $|z| \le c\alpha^{-1/2}$ for constant $c$?
I'm most interested in approximations for $G^{(\alpha)}_\ell(z) (1-z^2)^{\alpha/2}$ (which is what I need to bound certain inner products).
Any help greatly appreciated, Cris