an:02149552
Zbl 1062.33018
Terwilliger, Paul; Vidunas, Raimundas
Leonard pairs and the Askey-Wilson relations
EN
J. Algebra Appl. 3, No. 4, 411-426 (2004).
00113054
2004
j
33D45 05A30 15A03
Askey scheme; Askey-Wilson polynomial; \(q\)-Racah polynomial; Leonard pair; tridiagonal pair
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.
Stamatis Koumandos (Nicosia)