# Finite term relations for the exponential orthogonal polynomials

## TL;DR

This paper characterizes when exponential orthogonal polynomials satisfy finite-term relations, linking these conditions to geometric shapes, operator theory, and function theory, with implications for Hele-Shaw dynamics.

## Contribution

It establishes a precise criterion for finite-term relations in exponential orthogonal polynomials, connecting shape geometry, operator support, and Cauchy transforms.

## Key findings

- Three-term relation holds iff shape is an ellipse with uniform coloring.
- Finite support of the Hessenberg matrix's first row characterizes finite-term relations.
- Method to determine and reconstruct the shade function g from polynomial and Cauchy transform.

## Abstract

The exponential orthogonal polynomials encode via the theory of hyponormal operators a shade function $g$ supported by a bounded planar shape. We prove under natural regularity assumptions that these complex polynomials satisfy a three term relation if and only if the underlying shape is an ellipse carrying uniform black on white. More generally, we show that a finite term relation among these orthogonal polynomials holds if and only if the first row in the associated Hessenberg matrix has finite support. This rigidity phenomenon is in sharp contrast with the theory of classical complex orthogonal polynomials. On function theory side, we offer an effective way based on the Cauchy transforms of $g, \overline{z}g, \ldots, \overline{z}^dg$, to decide whether a $(d+2)$-term relation among the exponential orthogonal polynomials exists; in that case we indicate how the shade function $g$ can be reconstructed from a resulting polynomial of degree $d$ and the Cauchy transform of $g$. A discussion of the relevance of the main concepts in Hele-Shaw dynamics completes the article.