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 beta, gamma, gamma*,rho, rho*,omega, eta, eta* taken from K such that both A(2)A* - beta AA* A + A* A(2) - gamma(AA* + A*A) - rho A* = gamma* A(2) + omega A + eta I, A*(2) A - beta A* AA* + AA*(2) - gamma*(A*A + AA*) - 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.