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

Abstract

The sequence $\{{}_2{\varphi }_1(q^{-k},q^{b+1};q^{-\overline{b}-k+1};q,q^{-\overline{b}+1/2}z){\}}_{k\ge 0}$ 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 $\check{\omega }(b;\theta )=|(q{e}^{i\theta };q{)}_{\infty }/(q^b{e}^{i\theta };q{)}_{\infty }{|}^2$, where $0<q<1$ and $\Re (b)>0$. With the use of the basic hypergeometric polynomials ${}_2{\varphi }_1(q^{-k},q^b;q^{-\overline{b}-k+1};q,q^{-\overline{b}+1}z)$, $k\ge 0$, 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 Szegő functions are found.

References in corpus (2)

• Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas
• Extreme zeros in a sequence of para-orthogonal polynomials and bounds for the support of the measure

Cited by in corpus (2)

• Christoffel formula for kernel polynomials on the unit circle
• Orthogonal polynomials on the real line corresponding to a perturbed chain sequence