Two linear transformations each tridiagonal with respect to an eigenbasis of the other; an algebraic approach to the Askey scheme of orthogonal polynomials

Abstract

Let $K$ denote a field, and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V\to V$ and $A^{\ast }:V\to V$ that satisfy the following two conditions: There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^{\ast }$ is diagonal. There exists a basis for $V$ with respect to which the matrix representing $A^{\ast }$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. We give a correspondence between Leonard pairs and a class of orthogonal polynomials. This class coincides with the terminating branch of the Askey scheme and consists of the $q$-Racah, $q$-Hahn, dual $q$-Hahn, $q$-Krawtchouk, dual $q$-Krawtchouk, quantum $q$-Krawtchouk, affine $q$-Krawtchouk, Racah, Hahn, dual Hahn, Krawtchouk, Bannai/Ito, and orphan polynomials. We describe the above correspondence in detail. We show how, for the listed polynomials, the 3-term recurrence, difference equation, Askey-Wilson duality, and orthogonality can be expressed in a uniform and attractive manner using the corresponding Leonard pair. We give some examples that indicate how Leonard pairs arise in representation theory and algebraic combinatorics. We discuss a mild generalization of a Leonard pair called a tridiagonal pair. At the end we list some open problems. Throughout these notes our argument is elementary and uses only linear algebra. No prior exposure to the topic is assumed.

This revised version contains many new open problems. 83 pages. Lecture notes for the summer school on orthogonal polynomials and special functions, Universidad Carlos III de Madrid, Leganes, Spain. July 8--July 18, 2004. For more information see http://www.uc3m.es/uc3m/dpto/MATEM/summerschool/indice.html

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Cited by in corpus (30)

• A "missing" family of classical orthogonal polynomials
• Distance-regular graphs
• Orthogonal Polynomials from Hermitian Matrices
• Commutative association schemes
• Normalized Leonard pairs and Askey-Wilson relations
• Orthogonal Polynomials from Hermitian Matrices II
• Askey-Wilson relations and Leonard pairs
• Dunkl shift operators and Bannai-Ito polynomials
• The Erdős-Ko-Rado theorem for twisted Grassmann graphs
• Duality of q-polynomials, orthogonal on countable sets of points
• A bilinear form relating two Leonard systems
• Charting the $q$-Askey scheme
• Totally bipartite tridiagonal pairs
• The classification of Leonard triples of QRacah type
• Nonsymmetric Askey-Wilson polynomials and $Q$-polynomial distance-regular graphs
• Leonard pairs, spin models, and distance-regular graphs
• Integrable boundaries for the q-Hahn process
• Exactly and quasi-exactly solvable `discrete' quantum mechanics
• A characterization of Leonard pairs using the parameters $\{a_i{\}}_{i=0}^d$
• On the shape of a tridiagonal pair
• Leonard pairs having LB-TD form
• Leonard pairs having zero-diagonal TD-TD form
• The end-parameters of a Leonard pair
• Structure of Thin Irreducible Modules of a Q-polynomial Distance-Regular Graph
• The Erdős-Ko-Rado basis for a Leonard system
• Factorized $A_2$-Leonard pair
• Leonard pairs having specified end-entries
• Characterizations of asymptotic distributions of continuous-time Pólya processes
• Self-dual Leonard pairs
• Krawtchouk polynomials, the Lie algebra ${\mathfrak{sl}}_2$, and Leonard pairs