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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 This is no longer valid for general parameters \(\alpha,\beta \in \mathbb{C}\).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.

MSC:

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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