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

Consider the sequence of polesA = {α1, α2, . . .}, and suppose the rational functions φj with poles in A form an orthonormal system with respect to a Hermitian positive-definite inner product. Further, assume the φj satisfy a three-term recurrence relation. Let the rational function φ j\1 with poles in {α2, α3, . . .} represent the associated rational function of φj of order 1; i.e. the φ (1) j\1 do satisfy the same three-term recurrence relation as the φj . In this paper we then give a relation between φj 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 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.