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