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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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