Analytic characterizations of Jordan chains by pole cancellation functions of higher order

In this paper the analytic characterization of generalized poles of operator valued generalized Nevanlinna functions (including the length of Jordan chains of the representing relation) is completed. In particular, given a Jordan chain of the representing relation of length ℓ, we show that there exists a pole cancellation function of order at least ℓ, and, moreover, the construction shows that it is of surprisingly simple form.