  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
  $n\to\infty$.
