A Characterization of "Classical" d-Orthogonal Polynomials Original Research Article

We give a characterization of "classical" d-orthogonal polynomials through a vectorial functional equation. A sequence of monic polynomials {Bn}n ≥ 0 is called d-simultaneous orthogonal or simply d-orthogonal if it fulfils the following d + 1-st order recurrence relation: [formula] with the initial conditions [formula] Denoting by {Ln}n ≥ 0 the dual sequence of {Bn}n ≥ 0, defined by 〈Ln, Bm〉 = δn, m, n, m ≥ 0, then the sequence {Bn}n ≥ 0 is d-orthogonal if and only if [formula] for any integer α with 0 ≤ α ≤ d − 1. Now, the d-orthogonal sequence {Bn}n ≥ 0 is called "classical" if it satisfies the Hahn′s property, that is, the sequence {Qn}n ≥ 0 is also d-orthogonal where Qn(x) = (n + 1)− 1 B′n + 1(x), n ≥ 0 is the monic derivative. If Λ denotes the vector t(L0, L1, ..., Ld − 1), the main result is the following: the d-orthogonal sequence {Bn}n ≥ 0 is "classical" if and only if, there exist two d × d polynomial matrices Ψ = (ψν, μ), Φ = (φν, μ), deg ψν, μ ≤ 1, deg φν, μ ≤ 2 such that Ψ Λ + D(Ψ Λ) = 0 with conditions about regularity (see below). Moreover, some examples are given.