Biorthogonal rational functions of $R_{II}$ type

TL;DR

This paper constructs and proves biorthogonality of a new class of rational functions related to $R_{II}$ recurrence relations, introducing a novel Zhedanov method and a Christoffel type transformation for eigenvalue problems.

Contribution

It introduces a new biorthogonal rational function sequence of $R_{II}$ type and a Zhedanov method for proving biorthogonality, with applications to eigenvalue problem transformations.

Findings

Established biorthogonality of the rational functions.

Developed a Zhedanov method for proof.

Provided a Christoffel type transformation for eigenvalue problems.

Abstract

In this work, a sequence of orthonormal rational functions that is also biorthogonal to another sequence of rational functions arising from recurrence relations of $R_{II}$ type is constructed. The biorthogonality is proved by a procedure which we call Zhedanov method. A particular case of a sequence of orthonormal rational functions having denominators of special form is considered to motivate the general case. The particular case provides a Christoffel type transformation of the generalized eigenvalue problem with a reformulation different from the existing literature.