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The $q-$Onsager algebra and multivariable $q-$special functions

Pascal Baseilhac, Luc Vinet, Alexei Zhedanov

Abstract

Two sets of mutually commuting $q-$difference operators $x_i$ and $y_j$, $i,j=1, ...,N$ such that $x_i$ and $y_i$ generate a homomorphic image of the $q-$Onsager algebra for each $i$ are introduced. The common polynomial eigenfunctions of each set are found to be entangled product of elementary Pochhammer functions in $N$ variables and $N+3$ parameters. Under certain conditions on the parameters, they form two `dual' bases of polynomials in $N$ variables. The action of each operator with respect to its dual basis is block tridiagonal. The overlap coefficients between the two dual bases are expressed as entangled products of $q-$Racah polynomials and satisfy an orthogonality relation. The overlap coefficients between either one of these bases and the multivariable monomial basis are also considered. One obtains in this case entangled products of dual $q-$Krawtchouk polynomials. Finally, the `split' basis in which the two families of operators act as block bidiagonal matrices is also provided.

The $q-$Onsager algebra and multivariable $q-$special functions

Abstract

Two sets of mutually commuting difference operators and , such that and generate a homomorphic image of the Onsager algebra for each are introduced. The common polynomial eigenfunctions of each set are found to be entangled product of elementary Pochhammer functions in variables and parameters. Under certain conditions on the parameters, they form two `dual' bases of polynomials in variables. The action of each operator with respect to its dual basis is block tridiagonal. The overlap coefficients between the two dual bases are expressed as entangled products of Racah polynomials and satisfy an orthogonality relation. The overlap coefficients between either one of these bases and the multivariable monomial basis are also considered. One obtains in this case entangled products of dual Krawtchouk polynomials. Finally, the `split' basis in which the two families of operators act as block bidiagonal matrices is also provided.

Paper Structure

This paper contains 16 sections, 16 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. The $q-$Onsager algebra and $q-$difference operators
  4. Two 'dual' polynomial eigenbases
  5. Overlap coefficients and orthogonality
  6. The 'split' basis
  7. Concluding remarks
  8. $U_{q}(\widehat{sl_2})$, $U_{q}(sl_2)$ and polynomial bases
  9. The quantum algebra $U_{q}(\widehat{sl_2})$
  10. The evaluation representation of $U_{q}(\widehat{sl_2})$ (quantum loop algebra of $sl_2$) J2CP
  11. The $q-$difference operators realization of $U_q(sl_2)$
  12. The expansion coefficients of Example \ref{['ex2']}
  13. The $q-$Racah and the dual $q-$Krawtchouk polynomials
  14. The $q-$Racah polynomials:
  15. The dual $q-$Krawtchouk polynomials:
  16. ...and 1 more sections

Key Result

Proposition 2.1

Let $\{k_\pm,\epsilon_\pm\}$ be complex scalars. There is an algebra morphism $O_{q}(\widehat{sl_2}) \mapsto U_{q}(\widehat{sl_2})$ such that with

Theorems & Definitions (34)

  • Definition 2.1: Ter03Bas2
  • Remark 2.1
  • Proposition 2.1: Bas3
  • Proposition 2.2: BSh1
  • Remark 2.2
  • Proposition 2.3: See Bas3
  • Lemma 2.1
  • proof
  • Proposition 2.4
  • Proposition 3.1
  • ...and 24 more