A note on Sobolev orthogonality for Laguerre matrix polynomials

Let {L _n ^(A,λ) (x)} _n⩾0 be the sequence of monic Laguerre matrix polynomials defined on 0, ∞) by where A ∈ C ^r×r . It is known that {L _n ^(A,λ) (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 _n ^(A,λ) (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. Supported by the National Natural Science Foundation of China (No.10571122), the Beijing Natural Science Foundation (No.1052006), and the Project of Excellent Young Teachers and the Doctoral Programme Foundation of National Education Ministry of China.