Abstract
Around 2001 we classified the Leonard systems up to isomorphism. The proof was lengthy and involved considerable computation. In this paper we give a proof that is shorter and involves minimal computation. We also give a comprehensive description of the intersection numbers of a Leonard system.
Highlights
In the area of Algebraic Combinatorics there is an object called a commutative association scheme [2, 13]
This thesis helped to motivate the work of Bannai [2, p. i], who taught a series of graduate courses on commutative association schemes during September 1978–December 1982 at the Ohio State University
In [18, Appendix A] and [20, Sect. 19], a bijection is given between the isomorphism classes of Leonard systems over R, and the orthogonal polynomial systems that satisfy Askey-Wilson duality
Summary
In the area of Algebraic Combinatorics there is an object called a commutative association scheme [2, 13]. Assumption on d and explicitly describes all the limiting cases that show up This version gives a complete classification of the orthogonal polynomial systems that satisfy Askey-Wilson duality. 19], a bijection is given between the isomorphism classes of Leonard systems over R, and the orthogonal polynomial systems that satisfy Askey-Wilson duality. Appearing in [14] is the correspondence between Leonard pairs over R and the orthogonal polynomial systems that satisfy Askey-Wilson duality. Turning to the present paper, we obtain two main results: (i) an improved proof for the classification of Leonard systems; (ii) a comprehensive description of the intersection numbers of a Leonard system. Concerning our second main result, we mentioned earlier that the Leonard systems correspond to the orthogonal polynomial sequences that satisfy Askey-.
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