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Two linear transformations each tridiagonal with respect to an eigenbasis of the other. (English) Zbl 0980.05054
D. A. Douglas Leonard established in 1982 that the first and second eigenvalues of P- and Q-polynomial association schemes can be represented by Askey-Wilson polynomials; see Douglas A. Leonard [SIAM J. Math. Anal. 13, 656-663 (1982; Zbl 0495.33006)]. The mere statement of this important result takes 10 pages of the Bannai-Itô monograph; see E. Bannai and T. Itô [Algebraic combinatorics. I: Association schemes (1984; Zbl 0555.05019)]. By considering an algebraic axiomatization of the situation (Leonard systems) Terwilliger manages to clarify and simplify considerably this remarkable result.

MSC:
05E35 Orthogonal polynomials (combinatorics) (MSC2000)
05E30 Association schemes, strongly regular graphs
15A04 Linear transformations, semilinear transformations
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
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