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Bispectrality and the ad conditions

F. Alberto Grunbaum

Abstract

At the beginning of the study of the bispectral problem, see [18], the ad-conditions played a crucial role in finding non-classical instances. The connection with the ad-conditions has reappeared in several different incarnations of the bispectral problem. Here we show that, properly adapted versions of these conditions, see [44], can play an important role in areas including, for instance, the study of exceptional orthogonal polynomials. This is also true in the non-commutative case. Even at this more advanced stage of the field one may hope that finding explicit solutions of these ad-conditions will provide an additional route to new examples.

Bispectrality and the ad conditions

Abstract

At the beginning of the study of the bispectral problem, see [18], the ad-conditions played a crucial role in finding non-classical instances. The connection with the ad-conditions has reappeared in several different incarnations of the bispectral problem. Here we show that, properly adapted versions of these conditions, see [44], can play an important role in areas including, for instance, the study of exceptional orthogonal polynomials. This is also true in the non-commutative case. Even at this more advanced stage of the field one may hope that finding explicit solutions of these ad-conditions will provide an additional route to new examples.
Paper Structure (12 sections, 112 equations)

This paper contains 12 sections, 112 equations.

Table of Contents

  1. A brief introduction
  2. The contents of the paper
  3. A few examples of exceptional Hermite polynomials
  4. The case of Laguerre exceptional polynomials
  5. Solving some ad-conditions
  6. An explicit form of Reach's ad-conditions
  7. An explicit form for the new ad-conditions for some Hermite exceptional polynomials
  8. A peek at the non-commutative case
  9. The ad-conditions and the Heisenberg evolution of $\Theta(x)$
  10. Acquiring a degree of freedom after a step of the Darboux process
  11. Some remarks on the literature and some final thoughts
  12. Statements and Declarations