Generalized polynomial Bezoutian with respect to a Jacobson chain basis over an arbitrary field

We introduce a so-called generalized polynomial Bezoutian with respect to a Jacobson chain basis over an arbitrary field. Some characterization of this kind of matrix, such as the Barnett-type factorization and the intertwining relation with generalized hypercompanion matrix, are obtained. The diagonal reduction formula via the generalized confluent Vandermonde matrix similar to that of classical Bezoutian is presented. The method used is based on polynomial module and operator representation.