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Using quasi-Darboux transformations to construct exceptional matrix polynomials

Ignacio Bono Parisi, Antonio J. Durán, Ignacio N. Zurrián

TL;DR

This work addresses the challenge of constructing exceptional matrix polynomials in a noncommutative, matrix-valued setting. It introduces a central tool, the quasi-Darboux transformation, to generate eigenpolynomial sequences from a seed while overcoming noncommutativity, complemented by a non-Darboux construction for additional flexibility. The authors develop a general framework and provide five explicit examples across Hermite, Laguerre, and Gegenbauer types, including a weight with a Dirac delta, detailing the resulting weights, eigenstructures, and symmetry properties. They further observe high-order recurrence relations and multiple symmetry operators for the obtained families, underscoring novel features of the matrix case and pointing to open questions about weight existence, multi-step constructions, and recurrence structures.

Abstract

We introduce a couple of methods to construct exceptional matrix polynomials. One of them uses what we have called quasi-Darboux transformations. This seems to be a more powerful method to deal with the non-commutativity problems that appear when matrix-valued polynomials are considered. The other method does not use any transformation of Darboux type. Using both methods, we construct a collection of five illustrative examples that show how powerful our two methods are. The examples include exceptional matrix polynomials of Hermite, Laguerre, and Gegenbauer type, as well as an example with a weight matrix having a Dirac delta.

Using quasi-Darboux transformations to construct exceptional matrix polynomials

TL;DR

This work addresses the challenge of constructing exceptional matrix polynomials in a noncommutative, matrix-valued setting. It introduces a central tool, the quasi-Darboux transformation, to generate eigenpolynomial sequences from a seed while overcoming noncommutativity, complemented by a non-Darboux construction for additional flexibility. The authors develop a general framework and provide five explicit examples across Hermite, Laguerre, and Gegenbauer types, including a weight with a Dirac delta, detailing the resulting weights, eigenstructures, and symmetry properties. They further observe high-order recurrence relations and multiple symmetry operators for the obtained families, underscoring novel features of the matrix case and pointing to open questions about weight existence, multi-step constructions, and recurrence structures.

Abstract

We introduce a couple of methods to construct exceptional matrix polynomials. One of them uses what we have called quasi-Darboux transformations. This seems to be a more powerful method to deal with the non-commutativity problems that appear when matrix-valued polynomials are considered. The other method does not use any transformation of Darboux type. Using both methods, we construct a collection of five illustrative examples that show how powerful our two methods are. The examples include exceptional matrix polynomials of Hermite, Laguerre, and Gegenbauer type, as well as an example with a weight matrix having a Dirac delta.
Paper Structure (13 sections, 14 theorems, 226 equations)

This paper contains 13 sections, 14 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Orthogonal matrix polynomials
  4. Symmetric operators
  5. Exceptional matrix polynomials
  6. Darboux and quasi-Darboux transformation
  7. A collection of instructive examples
  8. First example: applying a one-step quasi-Darboux transformation with polynomial seed function to a Hermite type polynomials
  9. Second example: applying a two-step quasi-Darboux transformation with polynomial seed functions to a Hermite type polynomials
  10. Third example: applying a one-step quasi-Darboux transformation with non-polynomial seed function to a Laguerre type polynomials
  11. Fourth example: applying a one-step quasi-Darboux transformation with non-polynomial seed function to a Gegenbauer type polynomials
  12. Fifth example: constructing examples without using quasi-Darboux transformation
  13. Recurrence relations for the exceptional matrix polynomials

Key Result

Lemma 1

Assume that the matrix of functions $W(t)$ satisfies $W(a)=I$, for some real number $a$. Then $W$ reduces to scalar weights if and only if $W(t)W(s)=W(s)W(t)$ for all $t,s$ in the support of $W$.

Theorems & Definitions (37)

  • Definition 1
  • Lemma 1
  • Remark 1
  • Remark 2
  • Definition 2
  • Theorem 2
  • Definition 3
  • Definition 4
  • Lemma 3
  • proof
  • ...and 27 more