Normalized Leonard pairs and Askey–Wilson relations

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 relationsA2A∗-βAA∗A+A∗A2-γ(AA∗+A∗A)-ϱA∗=γ∗A2+ωA+ηI,A∗2A-βA∗AA∗+AA∗2-γ∗(A∗A+AA∗)-ϱ∗A=γA∗2+ωA∗+η∗I,for some scalars β, γ, γ∗, ϱ, ϱ∗, 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.