## Orthogonality of Jacobi polynomials with general parameters

A. B. J. Kuijlaars, A. Martínez-Finkelshtein, and R. Orive

### Abstract

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.

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### Key words

Jacobi polynomials, orthogonality, Rodrigues formula, zeros.

33C45.

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