On Laguerre-Sobolev matrix orthogonal polynomials

In this manuscript, we study some algebraic and differential properties of matrix orthogonal polynomials with respect to the Laguerre-Sobolev right sesquilinear form defined by ⟨ p , q ⟩ S ≔ ∫ 0 ∞ p * ( x ) W L A ( x ) q ( x ) d x + M ∫ 0 ∞ ( p ′ ( x ) ) * W ( x ) q ′ ( x ) d x , {\langle p,q\rangle }_{{\bf{S}}}:= \underset{0}{\overset{\infty }{\int }}{p}^{* }\left(x){{\bf{W}}}_{{\bf{L}}}^{{\bf{A}}}\left(x)q\left(x){\rm{d}}x+{\bf{M}}\underset{0}{\overset{\infty }{\int }}{(p^{\prime} \left(x))}^{* }{\bf{W}}\left(x)q^{\prime} \left(x){\rm{d}}x, where W L A ( x ) = e − λ x x A {{\bf{W}}}_{{\bf{L}}}^{{\bf{A}}}\left(x)={e}^{-\lambda x}{x}^{{\bf{A}}} is the Laguerre matrix weight, W {\bf{W}} is some matrix weight, p p and q q are the matrix polynomials, M {\bf{M}} and A {\bf{A}} are the matrices such that M {\bf{M}} is non-singular and A {\bf{A}} satisfies a spectral condition, and λ \lambda is a complex number with positive real part.