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.
Categories: math.CA
