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 $A^*:V\to V$ 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 $A^*$ is diagonal. There exists a basis for $V$ with respect to which the matrix representing $A^*$ 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 3term recurrence, difference equation, AskeyWilson 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 August 2004
 arXiv:
 arXiv:math/0408390
 Bibcode:
 2004math......8390T
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematical Physics;
 17B37
 EPrint:
 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 8July 18, 2004. For more information see http://www.uc3m.es/uc3m/dpto/MATEM/summerschool/indice.html