On Laguerre-Sobolev matrix orthogonal polynomials

Abstract

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)WLA(x)q(x)dx+M∫0∞(p′(x))*W(x)q′(x)dx,{\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 WLA(x)=e−λxxA{{\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, pp and qq 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.

Keywords

• matrix orthogonal polynomials
• laguerre-sobolev polynomials
• 33c45
• 42c05
• 46e99
• 34a30