An explicit example of polynomials orthogonal on the unit circle with a dense point spectrum generated by a geometric distribution

TL;DR

This paper introduces a new family of orthogonal polynomials on the unit circle with a dense spectrum, expressed via q-hypergeometric functions, and associated with a wrapped geometric distribution.

Contribution

It provides an explicit construction of orthogonal polynomials with dense spectra using q-hypergeometric functions and connects them to the wrapped geometric distribution.

Findings

Polynomials have a dense point spectrum on the unit circle.

Orthogonality measure is the wrapped geometric distribution.

Classical properties of these polynomials are analyzed.

Abstract

We present a new explicit family of polynomials orthogonal on the unit circle with a dense point spectrum. This family is expressed in terms of q-hypergeometric function of type ${_2}\phi_1$. The orthogonality measure is the wrapped geometric distribution. Some "classical" properties of the above polynomials are presented.