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

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, Carathéodory, and Nevanlinna functions. This transformation can be considered as an analog of the Szegő mapping but it is based on the Cayley transform, which relates the upper half-plane to the unit disk. 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−λJ)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.