Normalized Leonard pairs and Askey–Wilson relations

Abstract

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 relations A 2 A ∗ - β AA ∗ A + A ∗ A 2 - γ ( AA ∗ + A ∗ A ) - ϱ A ∗ = γ ∗ A 2 + ω A + η I , A ∗ 2 A - β A ∗ AA ∗ + AA ∗ 2 - γ ∗ ( A ∗ A + AA ∗ ) - ϱ ∗ A = γ A ∗ 2 + ω A ∗ + 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.