On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval (Q1270287)

In this paper, the author uses the relationship between orthogonal polynomials on the unit circle and the real line to derive interesting new results for the unit circle case. Consider the recurrence relation \[ S_{n+1}(x)= xS_n(x)-S_{n-1}(x), \] with \(S_0(x) :=1\), \(S_1(x)=x\) and \(u_n>0\) \(\forall n\). This generates a sequence \(\{S_n\}\) of monic polynomials that are orthogonal with respect to a symmetric measure \(\mu\) on a symmetric real interval \([-2c,2c]\): \[ \int^{2c}_{-2c}S_nS_md\mu=h_n\delta_{nm} (h_n>0). \] Fix a scaling parameter \(d>0\), and relate \(z\) and \(x\) by \(x=d(z^{1/2}+z^{-1/2})\). For \(n\geq 0\), let \[ P_n(z):={d^{-n-1}z^{n/2}\Bigl(z^{1/2}S_{n+1}(x)-{S_{n+1}(2d) \over S_n(2d)}S_n(x)\Bigr)\over z-1}. \] This is the DG (Delsarte-Genin) mapping of \(\{S_n\}\) with parameter \(d\). It is known that the \(\{P_n\}\) satisfy a recurrence of the form \[ P_{n+1}(z)=zP_n(z)-a_nP^*_n(z) \] where \(P_0(z)= P^*_0 (z)=1\), \(| a_n|<1\) and \(P^*_n(z): =z^n\overline {P_n(1/ \overline z})\). Consequently the \(\{P_n\}\) are orthogonal with respect to a measure on the unit circle: \[ \int^{2\pi}_0 P_n\overline {P_m}d\sigma =\kappa_n \delta_{mn} (\kappa_n >0). \] The author shows how different choices of \(d\) may be used to ``squeeze'' orthogonal polynomials on the circle, yielding new systems of orthogonal polynomials. Moreover, he shows how the DG mapping yields sieved polynomials, both on the circle and the interval, and also circle analogues of the Askey-Wilson polynomials. Finally, some \(q\)-orthogonal polynomials are discussed, as well as the relation of DG to chain sequences and to Bauer's \(g\)-algorithm.