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Bispectrality of the sieved Jacobi polynomials

Luc Vinet, Alexei Zhedanov

TL;DR

The authors establish bispectrality for the sieved Jacobi polynomials by first deriving a Dunkl-type first-order eigenoperator $L(N)$ for the CMV-Laurent polynomials associated to sieved Jacobi OPUC on the unit circle, then transferring the result to the real line via the Szegő map to obtain a second-order Dunkl operator eigenproblem for the sieved Jacobi polynomials on $\

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.

Bispectrality of the sieved Jacobi polynomials

TL;DR

The authors establish bispectrality for the sieved Jacobi polynomials by first deriving a Dunkl-type first-order eigenoperator for the CMV-Laurent polynomials associated to sieved Jacobi OPUC on the unit circle, then transferring the result to the real line via the Szegő map to obtain a second-order Dunkl operator eigenproblem for the sieved Jacobi polynomials on $\

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.
Paper Structure (11 sections, 7 theorems, 133 equations)

This paper contains 11 sections, 7 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. OPUC, CMV and Szegő map
  3. Jacobi OPUC and their eigenvalue problem
  4. Sieved Jacobi OPUC
  5. Algebraic relations
  6. Eigenvalue equations for the sieved Jacobi polynomials on the real line
  7. Commuting operators
  8. Special cases
  9. The generalized ultraspherical polynomials
  10. The sieved ultraspherical polynomials
  11. Conclusion

Key Result

Proposition 1

The CMV-Laurent polynomials $\psi_n(z;N)$ associated to the sieved Jacobi OPUC $\Phi_n(z;N)$ satisfy the eigenvalue equation where the operator $L(N)$ is with for even $N$ and for $N$ odd. The eigenvalues are

Theorems & Definitions (13)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • Proposition 4
  • Remark 6.1
  • Remark 6.2
  • Proposition 5
  • proof
  • ...and 3 more