Elliptic Orthogonal Polynomials and OPRL

• Victor Alves (Universidade de São Paulo)

In this presentation we introduce and study elliptic orthogonal polynomials (EOP), namely meromorphic functions on an elliptic curve that satisfy orthogonality conditions analogous to those of orthogonal polynomials. After briefly recalling standard properties of orthogonal polynomials in the real line (OPRL)—such as zero localization and interlacing—we investigate their counterparts in the elliptic setting. We then present a general decomposition theorem which, under suitable assumptions on the orthogonality contour, expresses EOP in terms of scalar polynomials satisfying associated orthogonality relations. This framework enables the explicit construction of families of elliptic orthogonal polynomials in terms of Jacobi polynomials. The presentation is based on joint work with Andrei Martínez-Finkelshtein.

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