Two linear transformations each tridiagonal with respect to an eigenbasis of the other: An Algebraic approach to the Askey scheme of orthogonal polynomials

By Paul M. Terwilliger. 2004

245 citations recorded by the source corpus.

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 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ca5d9fd32c2b49b73de61b15db64f2963c7280f1

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Citations in the corpus

A LIMIT q = −1 FOR THE BIG Q-JACOBI POLYNOMIALS (2012, methodology): “The polynomials Wn(x) satisfy the recurrence relation [2], [18] (5.”
The Rahman polynomials and the Lie algebra sl_3(C) (2010, methodology): “We view the actions of H and H̃ on V as a rank 2 generalization of a Leonard pair [12]; this is discussed along with some open problems at the end of the paper.”
A Lie-theoretic interpretation of multivariate hypergeometric polynomials (2011, background): “The Leonard pairs are defined and classified in [14].”
Tridiagonal pairs and the quantum affine algebra Uq(sl2) (代数的組合せ論) (2005, methodology): “A tridiagonal pair for which the Vi, V ∗ i all have dimension 1 is called a Leonard pair [11].”
Finite-Dimensional Irreducible Modules for the Three-Point 𝔰𝔩2 Loop Algebra (2007, background): “There is a close connection between the Leonard pairs and the orthogonal polynomials that make up the terminating branch of the Askey scheme [25], [35], [38].”
The classification of Leonard triples of QRacah type (2011, background): “By this and (4) the sequence (A − θi−1I) · · · (A − θ1I)(A − θ0I)v (0 i d) (8) is a basis for V .”
A ‘missing’ family of classical orthogonal polynomials (2010, background): “The pairs of operators (X, Y ), (Y, Z) and (Z,X) play the role of ‘Leonard pairs’ (see [16, 24]).”
Dunkl shift operators and Bannai-Ito polynomials (2011, background): “ive another explicit formula for the polynomials Pn(x) which will relate them to ”true” Wilson polynomials. Remark that the basis φn(x) is important in Terwilliger’s approach to Leonard duality [29], [30]. Similar bases exist for -1 Jacobi polynomials as well [33]. Equivalently, the basis φn(x) can ”