Representations of J-central J-Potapov functions in both nondegenerate and degenerate cases

Let J be an m×m signature matrix (i.e. J∗=J and J2=Im) and let D:={z∈C:|z|<1}. Denote PJ(D) the class of all J-Potapov functions in D, i.e. the set of all meromorphic m×m matrix-valued functions f in D with J-contractive values at all points of D at which f is holomorphic. Further, denote PJ,0(D) the subclass of all f∈PJ(D) which are holomorphic at the origin. Let f∈PJ,0(D), and let be the Taylor series representation of f in some neighborhood of 0. Then it was proved in [B. Fritzsche, B. Kirstein, U. Raabe, On the structure of J-Potapov sequences, Linear Algebra Appl., in press] that for each n∈N the matrix An can be described by its position in a matrix ball depending on the sequence . The J-Potapov function f is called J-central if there exists some k∈N such that for each integer j⩾k the matrix Aj coincides with the center of the corresponding matrix ball.In this paper, we derive left and right quotient representations of matrix polynomials for J-central J-Potapov functions in D. Moreover, we obtain recurrent formulas for the matrix polynomials involved in these quotient representations.