Weighted $L^2$-Norms of Gegenbauer polynomials

Abstract

We study integrals of the form \begin{equation*} \int_{-1}^1(C_n^{(λ)}(x))^2(1-x)^α(1+x)^β\, 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 $n\to \infty$.