Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas

TL;DR

This paper develops the theory of para-orthogonal polynomials on the unit circle, focusing on those satisfying three-term recurrence relations, extending previous results, and emphasizing the role of chain sequences.

Contribution

It provides a comprehensive theory for the para-orthogonal polynomials associated with the sum form, extending prior work and highlighting the importance of chain sequences.

Findings

Established three-term recurrence relations for the sum form of para-orthogonal polynomials.

Extended existing results to cover all nontrivial measures on the unit circle.

Provided examples and applications demonstrating the theory's utility.

Abstract

When a nontrivial measure $\mu$ on the unit circle satisfies the symmetry $d\mu(e^{i(2\pi-\theta)}) = - d\mu(e^{i\theta})$ then the associated OPUC, say $S_n$, are all real. In this case, Delsarte and Genin, in 1986, have shown that the two sequences of para-orthogonal polynomials $\{zS_{n}(z) + S_{n}^{\ast}(z)\}$ and $\{zS_{n}(z) - S_{n}^{\ast}(z)\}$ satisfy three term recurrence formulas and have also explored some further consequences of these sequences of polynomials such as their connections to sequences of orthogonal polynomials on the interval $[-1,1]$. The same authors, in (1988), have also provided a means to extend these results to cover any nontrivial measure on the unit circle. However, only recently in Costa, Felix and Sri Ranga (2013) and then in Castillo, Costa, Sri Ranga and Veronese (2014), the extension associated with the para-orthogonal polynomials $zS_{n}(z) - S_{n}^{\ast}(z)$ was thoroughly explored, especially from the point of view of the three term recurrence, and chain sequences play an important part in this exploration. The main objective of the present manuscript is to provide the theory surrounding the extension associated with the para-orthogonal polynomials $zS_{n}(z) + S_{n}^{\ast}(z)$ for any nontrivial measure on the unit circle. Like in Costa, Felix and Sri Ranga (2013) and Castillo, Costa, Sri Ranga and Veronese (2014), chain sequences also play an important role in this theory. Examples and applications are also provided to justify the results obtained.