A note on Sobolev orthogonality for Laguerre matrix polynomials
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Abstract
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).
$$L_n ^{(A,\lambda )} (x) = \frac{{n!}}{{( - \lambda )^n }}\sum\limits_{k = 0}^n {\frac{{( - \lambda )^k }}{{k!(n - k)!}}(A + I)_n [(A + I)_k ]^{ - 1} x^k } ,$$
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.
Key words
Laguerre matrix polynomial Sobolev orthogonality matrix moment functionalAMS (2000) subject classification
33C45 42C05 47A60Preview
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References
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