A CHARACTERIZATION OF CLASSICAL D-ORTHOGONAL POLYNOMIALS

We give a characterization of ''classical'' d-orthogonal polynomials through a vectorial functional equation. A sequence of monic polynomials {B-n}(n greater than or equal to 0) is called d-simultaneous orthogonal or simply d-orthogonal if it fulfils the following d + 1-st order recurrence relation: [GRAPHICS] gamma(m+1)(0) not equal 0, m greater than or equal to 0 with the initial conditions [GRAPHICS] Denoting by {L(n)}(n greater than or equal to 0) the dual sequence of {B-n)(n greater than or equal to 0), defined by (L(n), B-m,) = delta(n,) (m), n, m greater than or equal to 0, then the sequence {B-n}(n greater than or equal to 0) is d-orthogonal if and only if [GRAPHICS] for any integer alpha with 0 less than or equal to alpha less than or equal to d-1. Now, the d-orthogonal sequence {B-n}(n greater than or equal to 0) is called ''classical'' if it satisfies the Hahn's property, that is, the sequence {Q(n)}(n greater than or equal to 0) is also d-orthogonal where Q(n)(x)=(n + 1)(-1) B-n+1(t)(x), n greater than or equal to 0 is the monic derivative. If Lambda denotes the vector '(L(0), L(1), ..., L(d-1)), the main result is the following: the d-orthogonal sequence {B-n}(n greater than or equal to 0) is ''classical'' if and only if, there exist two d x d polynomial matrices Psi = (psi(nu,mu)), Phi = (phi(nu,mu)), deg psi(nu,mu) less than or equal to 1, deg phi(nu,mu) less than or equal to 2 such that Psi Lambda + D(Phi Lambda) = 0 with conditions about regularity (see below). Moreover, some examples are given. (C) 1995 Academic Press, Inc.