Recurrence relations and the Christoffel-Darboux formula for elliptic orthogonal polynomials

TL;DR

This paper explores elliptic orthogonal polynomials, deriving recurrence relations and a Christoffel-Darboux formula, revealing their deep connection to elliptic curves and simplifying under symmetry assumptions.

Contribution

It establishes fundamental recurrence relations and a Christoffel-Darboux formula for elliptic orthogonal polynomials, linking their coefficients to elliptic curve equations.

Findings

Derived five-term and seven-term recurrence relations.

Established a Christoffel-Darboux formula for elliptic polynomials.

Connected recurrence coefficients to elliptic curve equations.

Abstract

In recent years, there has been significant progress in the theory of orthogonal polynomials on algebraic curves, particularly on genus 1 surfaces. In this paper, we focus on elliptic orthogonal polynomials and establish several of their fundamental properties. In particular, we derive general five-term and seven-term recurrence relations, which lead to a Christoffel-Darboux formula and the construction of an associated point process on the A-cycle of the torus. Notably, the recurrence coefficients in these relations are intricately linked through the underlying elliptic curve equation. Under additional symmetry assumptions on the weight function, the structure simplifies considerably, recovering known results for orthogonal polynomials on the complex plane.