Associated Rational Functions based on a Three-term Recurrence Relation for Orthogonal Rational Functions∗
نویسندگان
چکیده
Consider the sequence of poles A = {α1, α2, . . .}, and suppose the rational functions φn with poles inA form an orthonormal system with respect to an Hermitian positive-definite inner product. Further, assume the φn satisfy a three-term recurrence relation. Let the rational function φ n\1 with poles in {α2, α3, . . .} represent the associated rational function of φn of order 1; i.e. the φ n\1 satisfy the same three-term recurrence relation as the φn. In this paper we then give a relation between φn and φ n\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 n\1 form an orthonormal system of rational functions with respect to an 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.
منابع مشابه
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...
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