## Affine transformations of a Leonard pair

### Summary

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.

### Mathematics Subject Classification

05E35, 05E30, 33C45, 33D45

### Keywords/Phrases

leonard pair, tridiagonal pair, q-racah polynomial, orthogonal polynomial