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→V and A*:V→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* is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A* 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 β,γ,γ*,ϱ,ϱ*,ω,η,η* taken from K such that both \begin{eqnarray*} A^2 A^*-\beta A A^*A+A^* A^2-\gamma (A A^*+A^*A) -\varrho\,A^* & = & \gamma^* A^2+\omega A+\eta\,I,\\[5pt] A^*{}^2A-\beta A^*AA^*+AA^*{}^2-\gamma^* (A^*A+A A^*)-\varrho^*A & = & \gamma A^*{}^2+\omega A^*+\eta^*I\,. \end{eqnarray*} The sequence is uniquely determined by the Leonard pair provided the dimension of V is at least...