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Bispectral rational functions and Leonard trios

Nicolas Crampé, Wolter Groenevelt, Quentin Labriet, Lucia Morey, Luc Vinet, Carel Wagenaar

TL;DR

The paper introduces Leonard trios as a natural extension of Leonard pairs and uses them to study bispectral rational functions. By linking two Leonard pairs through a shared operator $Z$ and employing algebraic Heun operators, it provides a framework in which overlap coefficients between eigenbases are biorthogonal rational functions, notably realizing Wilson rational functions. It then focuses on irreducible Leonard trios of $q$-Racah type, showing these overlaps coincide with Wilson functions and deriving their expressions in terms of $q$-Racah polynomials, along with explicit generalized eigenvalue problems and biorthogonal partners. Reduced Leonard trios are developed to capture $R_I$-type recurrences, including limits to dual $q$-Hahn and related cases. The work lays groundwork for a full bispectral rational-function scheme analogous to the Askey scheme and points to further algebraic and multivariate generalizations.

Abstract

It is well-known that Leonard pairs have a close connection with bispectral orthogonal polynomials of the Askey scheme. In this paper, we introduce the notion of a Leonard trio $(V,\oV,Z)$, an algebraic structure extending Leonard pairs, for which the overlap coefficients of eigenfunctions of $V$ and $\oV$ are biorthogonal rational functions satisfying generalized eigenvalue problems. We introduce and start the classification of irreducible Leonard trios by using its connection with Leonard pairs and Heun operators. In particular, we show that Wilson's rational functions appear as overlap coefficients, prove its difference, recurrence and biorthogonality relations, and obtain a summation formula expressing them as a finite sum of products of two $q$-Racah polynomials. We also begin to investigate reduced Leonard trios, for which the general eigenvalue problem simplifies to a $R_I$-type recurrence relation. As an illustration, we present an example of this in which the rational functions appearing as overlap coefficients can be expressed as a ${}_{4}φ_3$ and are associated with a Leonard pair of dual $q$-Hahn type.

Bispectral rational functions and Leonard trios

TL;DR

The paper introduces Leonard trios as a natural extension of Leonard pairs and uses them to study bispectral rational functions. By linking two Leonard pairs through a shared operator and employing algebraic Heun operators, it provides a framework in which overlap coefficients between eigenbases are biorthogonal rational functions, notably realizing Wilson rational functions. It then focuses on irreducible Leonard trios of -Racah type, showing these overlaps coincide with Wilson functions and deriving their expressions in terms of -Racah polynomials, along with explicit generalized eigenvalue problems and biorthogonal partners. Reduced Leonard trios are developed to capture -type recurrences, including limits to dual -Hahn and related cases. The work lays groundwork for a full bispectral rational-function scheme analogous to the Askey scheme and points to further algebraic and multivariate generalizations.

Abstract

It is well-known that Leonard pairs have a close connection with bispectral orthogonal polynomials of the Askey scheme. In this paper, we introduce the notion of a Leonard trio , an algebraic structure extending Leonard pairs, for which the overlap coefficients of eigenfunctions of and are biorthogonal rational functions satisfying generalized eigenvalue problems. We introduce and start the classification of irreducible Leonard trios by using its connection with Leonard pairs and Heun operators. In particular, we show that Wilson's rational functions appear as overlap coefficients, prove its difference, recurrence and biorthogonality relations, and obtain a summation formula expressing them as a finite sum of products of two -Racah polynomials. We also begin to investigate reduced Leonard trios, for which the general eigenvalue problem simplifies to a -type recurrence relation. As an illustration, we present an example of this in which the rational functions appearing as overlap coefficients can be expressed as a and are associated with a Leonard pair of dual -Hahn type.
Paper Structure (32 sections, 9 theorems, 134 equations)

This paper contains 32 sections, 9 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Leonard trio
  3. Definitions
  4. Bispectral rational functions
  5. Irreducible Leonard trio
  6. Definition and algebraic Heun operators
  7. Leonard pairs and bispectral polynomials
  8. Irreducible Leonard trios and bispectral polynomials
  9. Reduced Leonard trio
  10. Constraints for the eigenvalues of $V$ and $Z$ when $(V,Z)$ is a reduced LP
  11. Overlap coefficients for reduced LT
  12. Limits for LP of $q$-Racah type and reduced LP
  13. Irreducible Leonard trio of $q$-Racah type
  14. Constraints to obtain a LT
  15. Wilson rational functions and irreducible Leonard trio
  16. ...and 17 more sections

Key Result

Proposition 2.3

Let $(V,{\widetilde{V}},Z)$ be a LT. The functions defined by satisfy the recurrence relation and the difference equation where $X_{m,n}$ and $Z_{m,n}$ are the entries of the matrices ${\widetilde{V}} Z$ and $Z$ in the basis $\boldsymbol{v_n}$ and ${\widetilde{X}}_{x,y}$ and ${\widetilde{Z}}_{x,y}$ are the entries of the matrices $ZV$ and $Z$ in the basis $\boldsymbol{{\widetilde{v}}_x}$:

Theorems & Definitions (15)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • ...and 5 more