Askey-Wilson relations and Leonard pairs. (English) Zbl 1132.05061
Author’s abstract: It is known that if \((A,A^*)\) is a Leonard pair, then the linear transformations \(A, A^*\) satisfy the Askey-Wilson relations
\[ \begin{aligned} A^2 A^*-\beta AA^* A + A^* A^2- \gamma (AA^*+ A^*A) -\rho A^*&= \gamma^* A^2+ \omega A + \eta I,\\ A^{*2}A- \beta A^* AA^*+ AA^{*2}- \gamma^* (A^*A+AA^*)- \rho^*A &= \gamma A^{*2}+ \omega A^*+\eta^*I\end{aligned} \]
for some scalars \(\beta,\gamma,\gamma^*,\rho,\rho^*,\omega,\eta,\eta^*\). The problem of this paper is the following: given a pair of Askey-Wilson relations as above, how many Leonard pairs are there that satisfy those relations? It turns out that the answer is 5 in general. We give the generic number of Leonard pairs for each Askey-Wilson type of Askey-Wilson relations.
\[ \begin{aligned} A^2 A^*-\beta AA^* A + A^* A^2- \gamma (AA^*+ A^*A) -\rho A^*&= \gamma^* A^2+ \omega A + \eta I,\\ A^{*2}A- \beta A^* AA^*+ AA^{*2}- \gamma^* (A^*A+AA^*)- \rho^*A &= \gamma A^{*2}+ \omega A^*+\eta^*I\end{aligned} \]
for some scalars \(\beta,\gamma,\gamma^*,\rho,\rho^*,\omega,\eta,\eta^*\). The problem of this paper is the following: given a pair of Askey-Wilson relations as above, how many Leonard pairs are there that satisfy those relations? It turns out that the answer is 5 in general. We give the generic number of Leonard pairs for each Askey-Wilson type of Askey-Wilson relations.
Reviewer: P. R. Parthasarathy (Karlsruhe)
MSC:
| 05E35 | Orthogonal polynomials (combinatorics) (MSC2000) |
| 33D45 | Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
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