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.The research of A.B.J.K. and A.M.F. was partially supported by the European Research Network "NeCCA" INTAS 03-51-6637, by the Ministry of Science and Technology (MCYT) of Spain through grant BFM2001-3878-C02, and by NATO Collaborative Linkage Grant "Orthogonal Polynomials: Theory, Applications and Generalizations," ref. PST.CLG.979738. A.B.J.K. was also supported by FWO research projects G.0176.02 and G.0455.04, and by K.U. Leuven research grant OT/04/24. Additionally, A.M.F. acknowledges the support of Junta de Andalucía, Grupo de Investigación FQM 0229. The research of R.O. was partially supported by Research Projects of Spanish MCYT and Gobierno Autónomo de Canarias, under contracts BFM2001-3411 and PI2002/136, respectively