Askey–Wilson relations and Leonard pairs

It is known that if (A,A*)(A,A*) is a Leonard pair, then the linear transformations AA, A*A* satisfy the Askey–Wilson relationsA2A*-βAA*A+A*A2-γAA*+A*A-ϱA*=γ*A2+ωA+ηI,A*2A-βA*AA*+AA*2-γ*A*A+AA*-ϱ*A=γA*2+ωA*+η*Ifor some scalars β,γ,γ*,ϱ,ϱ*,ω,η,η*β,γ,γ*,ϱ,ϱ*,ω,η,η*. 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.