Normalized Leonard pairs and Askey-Wilson relations

Abstract

Let $V$ denote a vector space with finite positive dimension, and let $(A,B)$ denote a Leonard pair on $V$. As is known, the linear transformations $A,B$ satisfy the Askey-Wilson relations A^2B -bABA +BA^2 -g(AB+BA) -rB = hA^2 +wA +eI, B^2A -bBAB +AB^2 -h(AB+BA) -sA = gB^2 +wB +fI, for some scalars $b,g,h,r,s,w,e,f$. The scalar sequence is unique if the dimension of $V$ is at least 4. If $c,c\ast ,t,t\ast$ are scalars and $t,t\ast$ are not zero, then $(tA+c,t\ast B+c\ast )$ 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.

22 pages; corrected version, with improved presentation of Section 9

References in corpus (2)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• Two linear transformations each tridiagonal with respect to an eigenbasis of the other; an algebraic approach to the Askey scheme of orthogonal polynomials

Cited by in corpus (19)

• The Universal Askey-Wilson Algebra
• Askey-Wilson relations and Leonard pairs
• Towards a classification of the tridiagonal pairs
• A classification of sharp tridiagonal pairs
• An embedding of the Bannai-Ito algebra in $\mathcal{U}(\mathfrak{osp}(1,2))$ and $-1$ polynomials
• Sharp tridiagonal pairs
• The Clebsch--Gordan coefficients of $U({\mathfrak{sl}}_2)$ and the Terwilliger algebras of Johnson graphs
• Affine transformations of a Leonard pair
• The structure of a tridiagonal pair
• New realizations of algebras of the Askey-Wilson type in terms of Lie and quantum algebras
• Charting the $q$-Askey scheme. II. The $q$-Zhedanov scheme
• An Askey-Wilson Algebra of Rank 2
• Transition maps between the 24 bases for a Leonard pair
• On the shape of a tridiagonal pair
• The switching element for a Leonard pair
• Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair
• Finite-dimensional irreducible modules of the universal Askey-Wilson algebra
• The Clebsch-Gordan Rule for $U({\mathfrak{sl}}_2)$, the Krawtchouk Algebras and the Hamming Graphs
• Two non-nilpotent linear transformations that satisfy the cubic $q$-Serre relations