Representation-theoretic derivation of Verblunsky coefficients for sieved Jacobi OPUC

Derive explicit formulas for the Verblunsky coefficients of the sieved Jacobi orthogonal polynomials on the unit circle using only the representations of the generalized circle Jacobi algebra generated by the Dunkl-type operator L(N) together with the dihedral-group reflections Rj and rotations Tj, without relying on recurrence relations or mapping constructions.

Background

The paper introduces a generalized circle Jacobi algebra for the sieved Jacobi orthogonal polynomials on the unit circle, extending the ordinary circle Jacobi algebra by incorporating many reflections and rotations corresponding to the dihedral group DN. The authors establish anticommutation relations between the Dunkl-type operator L(N) and these reflection/rotation operators.

For the non-sieved (N=1) Jacobi OPUC, previous work showed that Verblunsky coefficients can be obtained purely from representations of the circle Jacobi algebra. Here, the authors suggest pursuing an analogous representation-theoretic derivation for the sieved case, which would provide an algebraic route to the explicit Verblunsky parameters independent of direct recurrence or mapping methods.

References

Deriving the explicit expressions of the Verblunsky coefficients of the sieved Jacobi OPUC from the construction of the representations of this generalized circle Jacobi algebra alone, is an interesting problem which we defer to future work.

— Bispectrality of the sieved Jacobi polynomials  (2501.12806 - Vinet et al., 22 Jan 2025) in Section 5 (Algebraic relations), final paragraph