Orthogonality of Jacobi polynomials with general parameters. (English) Zbl 1075.33005

Let \(P_{n}^{(\alpha,\beta)}(z)\) be the Jacobi polynomials of degree \(n\) and order \((\alpha,\beta)\). It is well-known that when \(\alpha,\beta>-1\), these polynomials are orthogonal on \([-1,1]\) with respect to the weight function \((1-z)^{\alpha}\,(1+z)^{\beta}\). As a result of the orthogonality, all their zeros are simple and belong to the interval \((-1,\,1)\). This is no longer valid for general parameters \(\alpha,\beta \in \mathbb{C}\). In this paper the authors show that for \(\alpha,\beta \in \mathbb{C}\), but excluding some special cases, the Jacobi polynomials \(P_{n}^{(\alpha,\beta)}(z)\) may still be characterized by orthogonality relations. In particular, they 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. The new orthogonality conditions obtained in this paper can be useful in the study of the zeros of the polynomials \(P_{n}^{(\alpha,\beta)}(z)\) when the parameters \(\alpha\) and \(\beta\) are not \(>-1\) and they also can be applied to establish asymptotic properties of these polynomials.


33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33F05 Numerical approximation and evaluation of special functions
33C90 Applications of hypergeometric functions
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