Weighted $L^2$-Norms of Gegenbauer polynomials

We study integrals of the form \begin{equation*} \int_{-1}^1(C_n^{(\lambda)}(x))^2(1-x)^\alpha (1+x)^\beta\, dx, \end{equation*} where $C_n^{(\lambda)}$ denotes the Gegenbauer-polynomial of index $\lambda>0$ and $\alpha,\beta>-1$. We give exact formulas for the integrals and their generating functions, and obtain asymptotic formulas as $n\to\infty$.