Askey-Wilson relations and Leonard pairs

Abstract

It is known that if $(A,B)$ is a Leonard pair, then the linear transformations $A$, $B$ satisfy the Askey-Wilson relations A^2 B - b A B A + B A^2 - g (A B+B A) - r B = h A^2 + w A + e I, B^2 A - b B A B + A B^2 - h (A B+B A) - s A = g B^2 + w B + f I, for some scalars $b,g,h,r,s,w,e,f$. The problem of this paper is the following: given a pair of Askey-Wilson relations as above, how many Leonard pairs are there that satisfy those relations? It turns out that the answer is 5 in general. We give the generic number of Leonard pairs for each Askey-Wilson type of Askey-Wilson relations.

22 pages; corrected version; the example of Section 2 has the normalization consistent with the rest of the paper

References in corpus (3)

• 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
• Normalized Leonard pairs and Askey-Wilson relations

Cited by in corpus (14)

• The Universal Askey-Wilson Algebra
• Bispectrality of the Complementary Bannai-Ito Polynomials
• Dunkl shift operators and Bannai-Ito polynomials
• Higher Rank Relations for the Askey-Wilson and $q$-Bannai-Ito Algebra
• Towards a classification of the tridiagonal pairs
• A classification of sharp tridiagonal pairs
• Sharp tridiagonal pairs
• 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
• Transition maps between the 24 bases for a Leonard pair
• On the shape of a tridiagonal pair
• The switching element for a Leonard pair
• Leonard pairs having zero-diagonal TD-TD form