Orthogonality of Jacobi polynomials with general parameters

In this paper we study the orthogonality conditions satisfied by Jacobi polynomials $P_n^{(\alpha ,\beta )}$ when the parameters $\alpha$ and $\beta$ are not necessarily $>-1$. We establish orthogonality on a generic closed contour on a Riemann surface. Depending on the parameters, this leads to either full orthogonality conditions on a single contour in the plane, or to multiple orthogonality conditions on a number of contours in the plane. In all cases we show that the orthogonality conditions characterize the Jacobi polynomial $P_n^{(\alpha ,\beta )}$ of degree $n$ up to a constant factor.