Leonard pairs and the Askey-Wilson relations

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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*,,,ω, η, η* taken from K such that both (i) A² A*-βA A*A+A*A²-γ(AA*+A*A) - A* =γ*A²+ωA+; (ii) A*²A-βA*AA*+AA*²-γ*(A*A+AA*) -A =γA*²+ωA*+η*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.