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 This is no longer valid for general parameters $$\alpha,\beta \in \mathbb{C}$$.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