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

Consider the sequence of poles A = {α_1,α_2, . . .}, and suppose the rational functions φ_j with poles in A form an orthonormal system with respect to a Hermitian positive-definite inner product. Further, assume the φ_j satisfy a three-term recurrence relation. Let the rational function φ^{(1)}_{j\1} with poles in {α_2, α_3, . . .} represent the associated rational function of φ_j of order 1; i.e. the φ_^{(1)}_{j\1} do satisfy the same three-term recurrence relation as the φ_j . In this paper we then give a relation between φ_j and φ^{(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 φ^{(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.