Weighted L2-norms of Gegenbauer polynomials

We study integrals of the form ∫-11(Cn(λ)(x))2(1-x)α(1+x)βdx,where Cn(λ) denotes the Gegenbauer-polynomial of index λ> 0 and α, β> - 1. We give exact formulas for the integrals and their generating functions, and obtain asymptotic formulas as n→ ∞.