I am currently trying to familiarize myself with the Gegenbauer polynomials $\left(C_n^{\nu}(x)\right)$ and was reading [L Caffarelli, A. Friedman, Partial Regularity of the zero-set of solutions of linear and super-linear elliptic equations (1985)].
In the paper, the authors have the following,
From [Volume 1 Higher Transcendental Functions (Volume I), pg 176 eqn (13)] we have \begin{equation} \frac{1}{2}\Gamma(\nu)^2 C_n^{\nu}(\cos \theta) = \sum_{j=0}^{[n/2]} \frac{\Gamma(j+\nu)\Gamma(n-j+\nu)\cos((n-2j)\theta)}{j!(n-j)!} \end{equation} and we then easily deduce that \begin{equation} \left\vert \frac{\mathrm d^k}{\mathrm dt^k}C_n^{\nu}(t) \right\vert \leq Cn^{2k+m-3}, \end{equation} for $k = 0,1,2$.
Note that $\Gamma (\nu)$ is the Gamma function and $m$ is the dimension in which we are working.
Also, in other papers I have seen $\nu = \frac{m-2}{2}$ but the authors above didn't do this.
I am not sure how the authors came to the conclusion above. I assume its just a simple calculation as they said but I cannot figure it out. Any help would be graciously appreciated.