On vector Hankel determinants

Abstract

Recently, a definition of Hankel determinants H k n whose entries belong to a real finite dimensional linear space R d has been given. This definition is based on designants and Clifford algebra. Such determinants appear in the theory of vector orthogonal polynomials, vector Padé approximants, in the algebraic approach to the vector ε-algorithm and other areas. Its fundamental algebraic property is that it is a vector of the real linear space R d . Sylvester's identity is still valid for computing recursively these determinants, involving elements of Clifford algebra. The aim of this paper is to show that this way (Sylverster's identity) is not an optimal one and to propose a more effecient alternative one, since it avoids the use of the Clifford algebra structure. This new identity will be also called Sylvester's identity since it is equivalent to the classical Sylvester's identity in the scalar case. It allows us also to recover the fundamental property more easily. Moreover, an expression of H k n in terms of classical determinants will be given and also some new determinantal identities.