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Ad-conditions for higher-order recurrences: Dunkl and matrix-valued settings

  • I. Bono Parisi
  • F. A. Grünbaum
  • I. Zurrián
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Abstract

The ad-conditions of Duistermaat and Grünbaum characterize bispectrality for a pair of differential operators. M.~Reach observed that a sequence of functions which is simultaneously an eigenfunction of a second-order differential operator and satisfies a recurrence relation of arbitrary order must satisfy a different kind of ad-conditions. We prove that the Reach ad-conditions are also sufficient, by an argument which is purely algebraic. This motivated us to explore ad-conditions in two new directions: to Dunkl differential-difference operators, where the coefficients of the ad-condition become elements of the Dunkl algebra and the reflection enters as a coefficient, and to matrix-valued orthogonal polynomials, where we exhibit ad-conditions for several families of exceptional matrix-valued polynomials satisfying five- and seven-term recurrence relations.

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Cite

BibTeXRIS
BibTeX
@misc{parisi2026adconditions,
  title = {{Ad-conditions for higher-order recurrences: Dunkl and matrix-valued settings}},
  author = {I. Bono Parisi and F. A. Grünbaum and I. Zurrián},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00219},
  eprint = {2611.00219},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00219},
}
RIS
TY  - UNPB
TI  - Ad-conditions for higher-order recurrences: Dunkl and matrix-valued settings
AU  - I. Bono Parisi
AU  - F. A. Grünbaum
AU  - I. Zurrián
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00219
UR  - https://arxiv.org/abs/2611.00219
AB  - The ad-conditions of Duistermaat and Grünbaum characterize bispectrality for a pair of differential operators. M.~Reach observed that a sequence of functions which is simultaneously an eigenfunction of a second-order differential operator and satisfies a recurrence relation of arbitrary order must satisfy a different kind of ad-conditions. We prove that the Reach ad-conditions are also sufficient, by an argument which is purely algebraic. This motivated us to explore ad-conditions in two new directions: to Dunkl differential-difference operators, where the coefficients of the ad-condition become elements of the Dunkl algebra and the reflection enters as a coefficient, and to matrix-valued orthogonal polynomials, where we exhibit ad-conditions for several families of exceptional matrix-valued polynomials satisfying five- and seven-term recurrence relations.
ER  -