Orthogonal rational functions, associated rational functions and functions of the second kind

Consider the sequence of poles A = {alpha(1), alpha(2),...}, and suppose the rational functions phi(j) with poles in A form an orthonormal system with respect to a Hermitian positive-definite inner product. Further, assume the phi(j) satisfy a three-term recurrence relation. Let the rational function phi((1))(j\1) with poles in {alpha(2), alpha(3),...} represent the associated rational function of phi(j) of order 1; i.e. the phi((1))(j\1) do satisfy the same three-term recurrence relation as the phi(j). In this paper we then give a relation between phi(j) and phi((1))(j\1) in terms of the so-called rational functions of the second kind. Next, under certain conditions on the poles in A, we prove that the phi((1))(j\1) form an orthonormal system of rational functions with respect to a Hermitian positive-definite inner product. Finally, we give a relation between associated rational functions of different order, independent of whether they form an orthonormal system.