Eigenvalue equations for sieved polynomials or proving Askey right again

2025/07/05 by Vinet, Luc, Zhedanov, Alexei
#33C45 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.03862

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.

Citations

• Bispectrality of the sieved Jacobi polynomials
• The CMV bispectrality of the Jacobi polynomials on the unit circle
• Bannai-Ito polynomials and dressing chains
• CMV matrices and little and big -1 Jacobi polynomials
• Dunkl shift operators and Bannai-Ito polynomials
• OPUC on One Foot
• Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array
• Two Linear Transformations each Tridiagonal with Respect to an Eigenbasis of the other; Comments on the Parameter Array
• Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle

Related

• Bispectrality of the sieved Jacobi polynomials
• A new approach to the theory of classical hypergeometric polynomials
• Jacobi operator, q-difference equation and orthogonal polynomials
• The Askey-Wilson polynomials and q-Sturm-Lioville problems
• On the affine analogue of Jack's and Macdonald's polynomials