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Leonard pairs and the $$q$$-Racah polynomials. (English) Zbl 1075.05090
Summary: Let $$\mathbb K$$ denote a field, and let $$V$$ denote a vector space over $$\mathbb K$$ 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: (i) There exists a basis for $$V$$ with respect to which the matrix representing $$A$$ is irreducible tridiagonal and the matrix representing $$A^*$$ is diagonal. (ii) There exists a basis for $$V$$ with respect to which the matrix representing $$A$$ is diagonal and the matrix representing $$A^*$$ is irreducible tridiagonal. We call such a pair a Leonard pair on $$V$$. In the appendix to [Linear Algebra Appl. 330, 149–203 (2001; Zbl 0980.05054)] we outlined a correspondence between Leonard pairs and a class of orthogonal polynomials consisting of the $$q$$-Racah polynomials and some related polynomials of the Askey scheme. We also outlined how, for the polynomials in this class, the 3-term recurrence, difference equation, Askey–Wilson duality, and orthogonality can be obtained in a uniform manner from the corresponding Leonard pair. The purpose of this paper is to provide proofs for the assertions which we made in that appendix.

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