A note on Sobolev orthogonality for Laguerre matrix polynomials

Abstract

Let {L ^(A,λ) _n (x)} _n⩾0 be the sequence of monic Laguerre matrix polynomials defined on [0, ∞) by

$$L_n ^{(A,\lambda )} (x) = \frac{{n!}}{{( - \lambda )^n }}\sum\limits_{k = 0}^n {\frac{{( - \lambda )^k }}{{k!(n - k)!}}(A + I)_n [(A + I)_k ]^{ - 1} x^k } ,$$

where A ∈ C^r×r. It is known that {L ^(A,λ) _n (x)} _n⩾0 is orthogonal with respect to a matrix moment functional when A satisfies the spectral condition that Re(z) > −1 for every z ∈ σ(A).

In this note we show that for A such that σ(A) does not contain negative integers, the Laguerre matrix polynomials L ^(A,λ) _n (x) are orthogonal with respect to a non-diagonal Sobolev-Laguerre matrix moment functional, which extends two cases: the above matrix case and the known scalar case.