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

摘要

Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V --> V and A* : V --> V satisfying both conditions below: (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. (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 beta, gamma, gamma*, rho, rho* taken from K such that both 0 = [A, A(2)A* - beta AA*A + A*A(2) - gamma (AA* + A*A) -rhoA*] 0 = [A*, A*(2)A - betaA*AA* + AA*(2) - gamma*(A*A + AA*) - rho *A] 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. (C) 2001 Elsevier Science Inc. All rights reserved. [References: 39]