Elliptic polynomials orthogonal on the unit circle with a dense point spectrum

TL;DR

This paper introduces explicit elliptic polynomials orthogonal on the unit circle with a dense point spectrum, expressing moments via Jacobi elliptic functions and connecting to elliptic hypergeometric functions.

Contribution

It provides new explicit examples of orthogonal polynomials on the unit circle with dense spectra, extending elliptic hypergeometric functions and generalizing Askey-Wilson polynomials.

Findings

Polynomials orthogonal on the unit circle with dense point spectrum.

Explicit expressions in terms of elliptic hypergeometric functions.

Connection to elliptic generalizations of classical orthogonal polynomials.

Abstract

We introduce two explicit examples of polynomials orthogonal on the unit circle. Moments and the reflection coefficients are expressed in terms of Jacobi elliptic functions. We find explicit expression for these polynomials in terms of a new type of elliptic hypergeometric function. We show that obtained polynomials are orthogonal on the unit circle with respect to a dense point meausure, i.e. the spectrum consists from infinite number points of increase which are dense on the unit circle. We construct also corresponding explicit systems of polynomials orthogonal on the interval of the real axis with respect to a dense point measure. They can be considered as an elliptic generalization of the Askey-Wilson polynomials of a special type.