Two families of orthogonal polynomials on the unit circle from basic hypergeometric functions

TL;DR

This paper introduces two new families of orthogonal polynomials on the unit circle derived from basic hypergeometric functions, providing explicit formulas for their properties and extending previous results.

Contribution

It develops explicit expressions for two parametric families of orthogonal polynomials on the unit circle using basic hypergeometric functions, expanding the class of known orthogonal polynomials.

Findings

Explicit formulas for the polynomials and their norms

Determination of Verblunsky coefficients

Expressions for Szegő functions

Abstract

The sequence $\{\,_2\phi_1(q^{-k},q^{b+1};\,q^{-\overline{b}-k+1};\, q, q^{-\overline{b}+1/2} z)\}_{k \geq 0}$ of basic hypergeometric polynomials is known to be orthogonal on the unit circle with respect to the weight function $|(q^{1/2}e^{i\theta};\,q)_{\infty}/(q^{b+1/2}e^{i\theta};\,q)_{\infty}|^2$. This result, where one must take the parameters $q$ and $b$ to be $0 < q < 1$ and $\Re(b) > -1/2$, is due to P.I. Pastro \cite{Pastro-1985}. In the present manuscript we deal with the orthogonal polynomials $\hat{\Phi}_{n}(b;.)$ and $\check{\Phi}_{n}(b;.)$ on the unit circle with respect to the two parametric families of weight functions $\hat{\omega}(b; \theta) = |(e^{i\theta};\,q)_{\infty}/(q^{b}e^{i\theta};\,q)_{\infty}|^2$ and $\check{\omega}(b;\theta) = |(qe^{i\theta};\,q)_{\infty}/(q^{b}e^{i\theta};\,q)_{\infty}|^2$, where $0 < q < 1$ and $\Re(b) > 0$. With the use of the basic hypergeometric polynomials $ _2\phi_1(q^{-k},q^{b};\,q^{-\overline{b}-k+1};\, q, q^{-\overline{b}+1} z)$, $k \geq 0$, which have zeros on the unit circle when $\Re(b) > 0$, simple expressions for the (monic) polynomials $\hat{\Phi}_{n}(b;.)$ and $\check{\Phi}_{n}(b;.)$, their norms, the associated Verblunsky coefficients and also the respective Szeg\H{o} functions are found.