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Meta Algebras and Special Functions: the Racah Case

Nicolas Crampé, Quentin Labriet, Lucia Morey, Satoshi Tsujimoto, Luc Vinet, Alexei Zhedanov

Abstract

Finite families of biorthogonal rational functions and orthogonal polynomials of Racah-type are studied within a unified algebraic framework based on the meta Racah algebra and its finite-dimensional representations. These functions are identified as overlap coefficients between eigensolutions of generalized and standard eigenvalue problems posited on the representation space. The approach naturally yields their orthogonality relations and bispectral properties.

Meta Algebras and Special Functions: the Racah Case

Abstract

Finite families of biorthogonal rational functions and orthogonal polynomials of Racah-type are studied within a unified algebraic framework based on the meta Racah algebra and its finite-dimensional representations. These functions are identified as overlap coefficients between eigensolutions of generalized and standard eigenvalue problems posited on the representation space. The approach naturally yields their orthogonality relations and bispectral properties.

Paper Structure

This paper contains 28 sections, 12 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. The Meta Racah Algebra
  3. Hahn algebra.
  4. Meta Hahn algebra.
  5. Meta $q$-Racah algebra.
  6. Racah algebra.
  7. Borel subalgebra of $\mathfrak{sl}_2$.
  8. Two-diagonal Representation
  9. (Generalized) Eigenbases
  10. Representations of $m{\mathfrak R}$ on various bases
  11. Representations in the $e$ and $e^*$ bases
  12. Representations in the $f$ and $f^*$ bases
  13. Representations in the $d$ and $d^*$ bases
  14. Link with the Leonard trios
  15. Racah Polynomials
  16. ...and 13 more sections

Key Result

Proposition 1

The most general algebraic Heun operator associated to the Leonard pair $( V, Z)$ acting bidiagonally as $Z$ is for $h_0,\, h_1,\, h_4 \in \mathbb{C}$.

Theorems & Definitions (30)

  • Definition 2.1
  • Remark 2.2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Definition 5.1
  • Proposition 3
  • proof
  • Proposition 4
  • ...and 20 more