Orthogonality of Jacobi polynomials with general parameters

Abstract

In this paper we study the orthogonality conditions satisfied by Jacobi polynomials Pn(α,β) when the parameters α and β 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 Pn(α,β) of degree n up to a constant factor.