Three-term recurrences and matrix orthogonal polynomials
Abstract
For the space of polynomials in a real variable x, but with coefficients which are r × r complex matrices, a prescribed sequence {Ωn}n≥0 of r × r matrices generates a functional ℒ by formula presented The definition of pseudo-orthogonality given for a polynomial family {Pn(cursive Greek chi)}n≥0 has ℒ[Pm(cursive Greek chi) , Pn(cursive Greek chi)] = 0 if m < n, for n ≥ 1, while that of orthogonality has ℒ[Pm(cursive Greek chi) , Pn(cursive Greek chi)] = 0 if m ≠ n; these are the same only in the case r = 1, where multiplication is commutative. For a given {Ωn}n≥0, a pseudo-orthogonal family exists iff the corresponding block Hankel matrices are non-singular. If {Pn(cursive Greek chi)}n≥0 is pseudo-orthogonal, then the leading coefficients of all its Pn(cursive Greek chi) must be non-singular; furthermore, it satisfies a three-term recurrence Pn+1(cursive Greek chi) = (Ancursive Greek chi + Bn) Pn(cursive Greek chi) - CnPn-1(cursive Greek chi) in which all An and Cn are non-singular. The converse of this latter, i.e. Favard's Theorem, but for matrix polynomials, holds. Three examples illustrate differing features, in a setting of Ω0 being Hermitian, or not.
