Weighted $$L^2$$ L 2 -norms of Gegenbauer polynomials

We study integrals of the form

$$begin{aligned} int _{-1}^1(C_n^{(lambda )}(x))^2(1-x)^alpha (1+x)^beta {{,mathrm{mathrm {d}},}}x, end{aligned}$$

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 (nrightarrow infty ).