Eigenvalue equations for sieved polynomials or proving Askey right again

TL;DR

This paper proves that sieved Jacobi polynomials are eigenfunctions of a Dunkl-type operator with cyclic reflections, confirming Askey's long-standing conjecture and deepening understanding of their algebraic structure.

Contribution

It establishes the existence of a specific Dunkl-type operator for sieved Jacobi polynomials, confirming Askey's conjecture and providing explicit operator construction.

Findings

Sieved Jacobi polynomials are eigenfunctions of a Dunkl-type operator.

The operator involves cyclic reflections related to roots of unity.

Confirms Askey's long-standing conjecture about these polynomials.

Abstract

The sieved Jacobi polynomials have been introduced by Askey. These can be obtained from conveniently taking $q$ to be a root of unity in the Askey-Wilson polynomials. The question of determining if they are eigenfunctions of some operator has been lingering for a long time. Askey impressed on us his conviction that it had an affirmative answer. It is shown that he was right and that this operator is of Dunkl type with cyclic reflections corresponding to the powers of $q$.