Sieved Jacobi polynomials are eigenfunctions of an explicit Dunkl-type operator with cyclic reflections, confirming Askey's conjecture and establishing their bispectrality.
Bispectrality of the sieved Jacobi polynomials
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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.
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Eigenvalue equations for sieved polynomials or proving Askey right again
Sieved Jacobi polynomials are eigenfunctions of an explicit Dunkl-type operator with cyclic reflections, confirming Askey's conjecture and establishing their bispectrality.