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THE CHRISTOFFEL-DARBOUX IDENTITY FOR MATRIX ORTHONORMAL POLYNOMIALS
AM´ILCAR BRANQUINHO
Dedicated to Professor J. A. Sampaio Martins on the occasion of his sixtieth birthday
Abstract. In this work we show that the Christoffel-Darboux identity and its confluent form characterize the sequences of matrix orthonormal polynomials.
1. Introduction
In 1990, Brezinski established (see [3]) the following result concerning the Chris-
toffel-Darboux identity for the scalar sequences of orthonormal polynomials, {pn}.
Theorem (Brezinski). Let {pn} be a sequence of polynomials, such that for each n ∈ N = {0,1,...}, degpn = n. A necessary and sufficient condition in order to {pn} be a orthonormal polynomial sequence is, there exists a sequence (an) ⊂ R \ {0} such that
n
an+1(pn+1(x)pn(y) − pn(x)pn+1(y)) = (x − y) pk(x)pk(y) , n ∈ N ,
k=0
i.e. {pn} verifies a Christoffel-Darboux identity.
Note that, by Favard’s theorem, {pn} is an orthonormal polynomial sequence if, and only if there exists sequences (bn) ⊂ R and (an) ⊂ R \ {0} such that p−1(x) = 0, p0(x) = 1, and
xpn(x) = an+1pn+1(x) + bnpn(x) + anpn−1(x) , n ∈ N .
The purpose of this work is to extend Christoffel-Darboux identity to sequences
of orthonormal matrix polynomials.
2000 Mathematics Subject Classification. 42C05.
Key words and phrases. Orthogonal polynomials; Matrix polynomials.
This work was supported by Funda¸ca˜o para a Ciˆencia e Tecnologia SFRH/BPD/5600/2001
and Centro de Matem´atica da Universidade de Coimbra.
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