$R_{II}$ type recurrence, generalized eigenvalue problem and orthogonal polynomials on the unit circle

TL;DR

This paper explores a special recurrence relation for polynomials with real simple zeros, linking them to generalized eigenvalue problems and orthogonality on the unit circle, with theoretical insights and illustrative examples.

Contribution

It establishes a connection between $R_{II}$ type recurrence relations, eigenvalue problems, and orthogonal polynomials on the unit circle, providing new theoretical insights.

Findings

Polynomials satisfy a specific $R_{II}$ recurrence with real simple zeros.

Each polynomial $P_n$ is the characteristic polynomial of a generalized eigenvalue problem.

A positive measure on the unit circle associated with the recurrence is constructed.

Abstract

We consider a sequence of polynomials $\{P_n\}_{n \geq 0}$ satisfying a special $R_{II}$ type recurrence relation where the zeros of $P_n$ are simple and lie on the real line. It turns out that the polynomial $P_n$, for any $n \geq 2$, is the characteristic polynomial of a simple $n \times n$ generalized eigenvalue problem. It is shown that with this $R_{II}$ type recurrence relation one can always associate a positive measure on the unit circle. The orthogonality property satisfied by $P_n$ with respect to this measure is also obtained. Finally, examples are given to justify the results.