Normalized Leonard pairs and Askey–Wilson relations Original Research Article

Let V denote a vector space with finite positive dimension, and let (A, A*) denote a Leonard pair on V. As is known, the linear transformations A, A* satisfy the Askey–Wilson relationsimagefor some scalars β, γ, γ*, varrho, varrho*, ω, η, η*. The scalar sequence is unique if the dimension of V is at least 4. If c, c*, t, t* are scalars and t, t* are not zero, then (tA + c, t*A* + c*) is a Leonard pair on V as well. These affine transformations can be used to bring the Leonard pair or its Askey–Wilson relations into a convenient form. This paper presents convenient normalizations of Leonard pairs by the affine transformations, and exhibits explicit Askey–Wilson relations satisfied by them.