Geodesic
Affine transformations of a Leonard pair
The electronic journal of linear algebra, Tome 16 (2007), pp. 389-417.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Let K denote a field and let V denote a vector space over K with finite positive dimension. An ordered pair is considered of linear transformations A : V ! V and A* : V ! V that satisfy (i) and (ii) below: (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 irreducible tridiagonal and the matrix representing A is diagonal. Such a pair is called a Leonard pair on V . Let ,, i, ,*, i* denote scalars in K with ,, ,* nonzero, and note that ,A + iI, ,*A* + i*I is a Leonard pair on V . Necessary and sufficient conditions are given for this Leonard pair to be isomorphic to A, A*. Also given are necessary and sufficient conditions for this Leonard pair to be isomorphic to the Leonard pair A*, A.
Classification : 05E35, 05E30, 33C45, 33D45
Keywords: leonard pair, tridiagonal pair, q-racah polynomial, orthogonal polynomial 
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     title = {Affine transformations of a {Leonard} pair},
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     year = {2007},
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Nomura, Kazumasa; Terwilliger, Paul. Affine transformations of a Leonard pair. The electronic journal of linear algebra, Tome 16 (2007), pp. 389-417. http://geodesic.mathdoc.fr/item/ELA_2007__16__a5/