Bispectrality of the sieved Jacobi polynomials

TL;DR

This paper demonstrates that sieved Jacobi polynomials are bispectral, satisfying eigenvalue equations with Dunkl-type differential operators on both the unit circle and the real line, revealing new bispectral properties.

Contribution

It establishes the bispectrality of sieved Jacobi polynomials and derives associated eigenvalue equations involving Dunkl operators, extending known results to these polynomials.

Findings

Sieved Jacobi polynomials satisfy eigenvalue equations with Dunkl operators.

Eigenfunctions of second-order Dunkl operators on the real line.

Special cases include ultraspherical polynomials of the first and second kind.

Abstract

It is shown that the CMV Laurent polynomials associated to the sieved Jacobi polynomials on the unit circle satisfy an eigenvalue equation with respect to a first order differential operator of Dunkl type. Using this result, the sieved Jacobi polynomials on the real line are found to be eigenfunctions of a Dunkl differential operator of second order. Eigenvalue equations for the sieved ultraspherical polynomials of the first and second kind are obtained as special cases. These results mean that the sieved Jacobi polynomials (either on the unit circle or on the real line) are bispectral.