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Leonard pairs and the Askey-Wilson relations. (English) Zbl 1062.33018
Let $$\mathbb{K}$$ denote a field and let $$V$$ denote a vector space over $$\mathbb{K}$$ with finite positive dimension. An ordered pair of linear transformations $$A:\, V \rightarrow V$$ and $$A^{*}:\, V \rightarrow V$$ which satisfy the following two properties: (i) There exists a basis for $$V$$ with respect to which the matrix representing $$A$$ is irreducible tri-diagonal and the matrix representing $$A^{*}$$ is diagonal, (ii) There exists a basis for $$V$$ with respect to which the matrix representing $$A^{*}$$ is irreducible tri-diagonal and the matrix representing $$A$$ is diagonal, is called a Leonard pair on $$V$$. The authors show that there exists a sequence of scalars $$\beta, \,\gamma,\,\gamma^{*},\,\rho,\, \rho^{*},\, \omega,\, \eta,\, \eta^{*}$$ taken from $$\mathbb{K}$$ such that both $A^2A^{*}-\beta AA^{*}A+A^{*}A^2-\gamma\,(AA^{*}+A^{*}A)-\rho \,A^{*}=\gamma^{*}A^2+\omega A+\eta \,I,$ and $A^{* 2}A-\beta A^{*}A A^{*}+AA^{* 2}-\gamma^{*}\,(A^{*}A+A A^{*})-\rho^{*}A=\gamma A^{* 2}+\omega A^{*}+\eta^{*}\, I.$ The sequence is uniquely determined by the Leonard pair provided the dimension of $$V$$ is at least 4. The equations above are called the Askey-Wilson relations.

##### MSC:
 33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) 05A30 $$q$$-calculus and related topics 15A03 Vector spaces, linear dependence, rank, lineability
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