Typically the orthogonality relation for the Gegenbauer polynomials is given as:
$ \int_{-1}^{1}C_{n}^{\alpha}(x)C_{m}^{\alpha}(x)\cdot(1-x^2)^{\alpha-1/2}dx=\frac{\pi2^{1-2\alpha}\Gamma(2\alpha+n)}{n!(n+\alpha)(\Gamma(\alpha))^2}\delta_{mn} $
The Rodrigues formula for $C_{n}^{\alpha}(x)$ is given by:
$ C_{n}^{\alpha}(x)=constant\cdot(1-x^2)^{-\alpha+1/2}\frac{d^n}{dx^n}\left[(1-x^2)^{n+\alpha-1/2}\right] $
My question is the following:
How do I derive/proove the orthogonality relation?
In George Andrews' "Special Functions" Ch. 6, he proves the orthogonality relation for the Hermite and Laguerre polynomials by substituting the Rodrigues form of the polynomial in for one of the polynomials and then using integration by parts $n$ times. However when I try that I get something like:
$ \sum_{k=1}^{m}(-1)^{k+1}C_{n-(k-1)}^{\alpha+(k-1)}(x)\frac{d^{m-k}}{dx^{m-k}}\left[(1-x^2)^{m+\alpha-1/2}\right]\bigg|_{-1}^{1}+\int_{-1}^{1}C_{n-m}^{\alpha+m}(x)\cdot(1-x^2)^{m+\alpha-1/2}dx $
I don't immediately see why all the terms in the summation go to zero, nor how to evaluate the integral directly. Any ideas?