Leonard pairs and the Askey-Wilson relations

Let K denote a field and let $V$ denote a vector space over K with finite positive dimension. We consider an ordered pair of linear transformations $A:V\to V$ and $A^{\ast }:V\to V$ which satisfy the following two properties: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^{\ast }$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^{\ast }$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. Referring to the above Leonard pair, we show there exists a sequence of scalars $\beta,\gamma,\gamma^, \varrho,\varrho^,\omega, \eta, \eta^$ taken from K such that both (i) A^2 A^-\beta A A^*A+A^A^2-\gamma (AA^+A^A) -\varrho A^ =\gamma^*A^2+\omega A+\etaI; (ii) A^{*2}A-\beta A^AA^+AA^{2}-\gamma^(A^A+AA^) -\varrho^*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.