Basic hypergeometric functions and orthogonal Laurent polynomials (Q2884428)

Using a three-term contiguous relation satisfied by \(q\)-hypergeometric functions, the authors define and study recurrence coefficients, moments, orthogonality and asymptotic properties for the three-parameter class of orthogonal Laurent polynomials \(\displaystyle\left\{z^{\left\lfloor n/2\right\rfloor} Q^{(b,c,d)}_{n}(z)\right\}^{\infty}_{n=0}\) on the unit circle \(\displaystyle\mathcal{C}=\left\{z=e^{i\theta};~0<\theta<2\pi\right\}\), where the monic polynomials \(Q^{(b,c,d)}_{n}(z)\), \(n\geq0\), are given by NEWLINENEWLINE\[NEWLINEQ^{(b,c,d)}_{n}(z)=\frac{(q^{c-b+1};q)_n}{(q^{b+1};q)_n}~q^{n(b-d+1)}~_{2}\Phi_{1}(q^{-n}, q^{b+1}; q^{-c+b-n}; q, q^{-c+d-1}z),NEWLINE\]NEWLINE where \(0<q<1\) and the three complex parameters \(b\), \(c\) and \(d\) are such that \(b\neq -1, -2, \dots\), \(c-b+1\neq -1, -2, \dots\), \(\operatorname{Re}(d)>0\) and \(\operatorname{Re}(c+2-d)>0\). By a special choice of the parameters, results regarding a class of Szegő polynomials are also derived in the concluding section.NEWLINENEWLINEThe paper is a good contribution to the further development of \(q\)-polynomials and may find applications in mathematics and physics.