Elliptic orthogonal polynomials and OPRL

TL;DR

This paper introduces elliptic orthogonal polynomials on genus one Riemann surfaces, extending classical orthogonal polynomials to elliptic curves and exploring their properties, recurrence relations, and zero interlacing phenomena.

Contribution

It develops the theory of elliptic orthogonal a-polynomials, including their construction, recurrence, Christoffel--Darboux formula, and connections to classical orthogonal polynomials, with new interlacing phenomena.

Findings

Elliptic orthogonal polynomials exhibit zero interlacing similar to classical OPRL.

Established recurrence relations and Christoffel--Darboux formula for elliptic orthogonal functions.

Identified new interlacing phenomena from rational deformations of weights.

Abstract

We explore a class of meromorphic functions on elliptic curves, termed \emph{elliptic orthogonal a-polynomials} ($a$-EOPs), which extend the classical notion of orthogonal polynomials to compact Riemann surfaces of genus one. Building on Bertola's construction of orthogonal sections, we study these functions via non-Hermitian orthogonality on the torus, establish their recurrence properties, and derive an analogue of the Christoffel--Darboux formula. We demonstrate that, under real-valued orthogonality conditions, $a$-EOPs exhibit interlacing and simplicity of zeros similar to orthogonal polynomials on the real line (OPRL). Furthermore, we construct a general correspondence between families of OPRL and elliptic orthogonal functions, including a decomposition into multiple orthogonality relations, and identify new interlacing phenomena induced by rational deformations of the orthogonality weight.