Two linear transformations each tridiagonal with respect to an eigenbasis of the other Original Research Article

Let image denote a field, and let V denote a vector space over image with finite positive dimension. We consider a pair of linear transformations A:V→V and A*:V→V satisfying both conditions below: 1. [(i)] There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A* is irreducible tridiagonal. 2. [(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. Refining this notion a bit, we introduce the concept of a Leonard system. We give a complete classification of Leonard systems. Integral to our proof is the following result. We show that for any Leonard pair A,A* on V, there exists a sequence of scalars β,γ,γ*,varrho,varrho* taken from image such that bothimage where [r,s] means rs−sr. The sequence is uniquely determined by the Leonard pair if the dimension of V is at least 4. We conclude by showing how Leonard systems correspond to q-Racah and related polynomials from the Askey scheme.