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Meta algebras and biorthogonal rational functions: the $q$-Hahn case

Pierre-Antoine Bernard, Abderahmane Bouziane, Samuel Pellerin, Simone Têtu, Satoshi Tsujimoto, Luc Vinet, Meri Zaimi, Alexei Zhedanov

Abstract

A unified algebraic interpretation of both finite families of orthogonal polynomials and biorthogonal rational functions of $q$-Hahn type is provided. The approach relies on the meta $q$-Hahn algebra and its finite-dimensional bidiagonal representations. The functions of $q$-Hahn type are identified as overlaps (up to global factors) between bases solving ordinary or generalized eigenvalue problems in the representation of the meta $q$-Hahn algebra. Moreover, (bi)orthogonality relations, recurrence relations, difference equations and some contiguity relations satisfied by these functions are recovered algebraically using the actions of the generators of the meta $q$-Hahn algebra on various bases.

Meta algebras and biorthogonal rational functions: the $q$-Hahn case

Abstract

A unified algebraic interpretation of both finite families of orthogonal polynomials and biorthogonal rational functions of -Hahn type is provided. The approach relies on the meta -Hahn algebra and its finite-dimensional bidiagonal representations. The functions of -Hahn type are identified as overlaps (up to global factors) between bases solving ordinary or generalized eigenvalue problems in the representation of the meta -Hahn algebra. Moreover, (bi)orthogonality relations, recurrence relations, difference equations and some contiguity relations satisfied by these functions are recovered algebraically using the actions of the generators of the meta -Hahn algebra on various bases.

Paper Structure

This paper contains 25 sections, 10 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Meta $q$-Hahn algebra
  3. Bidiagonal finite-dimensional representations
  4. (Generalized) Eigenbases
  5. Representations of $m\mathfrak{h}_q$ in various bases
  6. Representation in the $e$ and $e^{*}$ bases.
  7. Representation in the $f$ and $f^{*}$ bases.
  8. $q$-Hahn and dual $q$-Hahn polynomials
  9. Identification
  10. Orthogonality relations
  11. Bispectral properties
  12. Recurrence relation
  13. Difference equation
  14. $q$-Hahn rational functions
  15. Representation theoretic interpretation
  16. ...and 10 more sections

Key Result

Proposition 3.1

There is a bidiagonal representation of the meta $q$-Hahn algebra $m\mathfrak{h}_q$ on the $N+1$-dimensional vector space $\mathcal{V}_N$ given by the following actions of the generators $Z,X,V$ on the basis vectors $\ket{n}$, for $n=0,1,\dots,N$: where $a_n$ are constants related to the normalization of the basis vectors, with $a_{-1}=a_N=0$, and $\alpha,\beta$ are two parameters related as foll

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Proposition 4.1
  • proof
  • Proposition 6.1
  • proof
  • ...and 16 more