A note on Wall's modification of the Schur algorithm and linear pencils of Jacobi matrices

TL;DR

This paper revisits Wall's transformation linking continued fractions of special functions to orthogonal rational functions, revealing new relations with Jacobi matrices and pseudo-Jacobi polynomials.

Contribution

It introduces a Wall transformation-based framework connecting continued fractions, orthogonal rational functions, and Jacobi matrices, extending classical results to rational functions.

Findings

Transformation establishes a one-to-one correspondence between continued fractions and orthogonal rational functions.

The recurrence relation involves Hermitian Jacobi matrices with a version of the Denisov-Rakhmanov theorem.

Pseudo-Jacobi polynomials are incorporated into the framework, illustrating broader applicability.

Abstract

In this note we revive a transformation that was introduced by H. S. Wall and that establishes a one-to-one correspondence between continued fraction representations of Schur, Carath\'eodory, and Nevanlinna functions. This transformation can be considered as an analog of the Szeg\H{o} mapping but it is based on the Cayley transform, which relates the upper half-plane to the unit disc. For example, it will be shown that, when applying the Wall transformation, instead of OPRL, we get a sequence of orthogonal rational functions that satisfy three-term recurrence relation of the form $(H-\lambda J)u=0$, where $u$ is a semi-infinite vector, whose entries are the rational functions. Besides, $J$ and $H$ are Hermitian Jacobi matrices for which a version of the Denisov-Rakhmanov theorem holds true. Finally we will demonstrate how pseudo-Jacobi polynomials (aka Routh-Romanovski polynomials) fit into the picture.