Affine transformations of a Leonard pair

Let K denote a field and let V denote a vector space over K withfinite positivedimension. An ordered pair is considered of linear transformations A : V → V and A∗ : V → V thatsatisfy (i) and (ii) below: (i) There exists a basis for V withrespect to whichthe matrix representing A is irreducibletridiagonal and the matrix representing A∗ is diagonal. (ii) There exists a basis for V withrespect to whichthe matrix representing A∗ is irreducibletridiagonal and the matrix representing A is diagonal. Sucha pair is called a Leonard pair on V . Let ξ, ζ, ξ∗, ζ∗ denote scalars in K with ξ, ξ∗ nonzero, andnote that ξA + ζI, ξ∗A∗ + ζ∗I is a Leonard pair on V . Necessary and sufficient conditions are givenfor this Leonard pair to be isomorphic to A, A∗. Also given are necessary and sufficient conditionsfor this Leonard pair to be isomorphic to the Leonard pair A∗, A.