Askey-Wilson relations and Leonard pairs

It is known that if (A, A*) is a Leonard pair, then the linear transformations A, A* satisfy the Askey-Wilson relations 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 for some scalars beta, gamma, gamma*, rho, rho*, omega, eta, eta*. The problem of this paper is the following: given a pair of Askey-Wilson relations as above, how many Leonard pairs are there that satisfy those relations? It turns out that the answer is 5 in general. We give the generic number of Leonard pairs for each Askey-Wilson type of Askey-Wilson relations. (C) 2007 Elsevier B.V. All rights reserved.