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 onetoone 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 halfplane 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 threeterm recurrence relation of the form $(H\lambda J)u=0$, where $u$ is a semiinfinite vector, whose entries are the rational functions. Besides, $J$ and $H$ are Hermitian Jacobi matrices for which a version of the DenisovRakhmanov theorem holds true. Finally we will demonstrate how pseudoJacobi polynomials (aka RouthRomanovski polynomials) fit into the picture.
 Publication:

arXiv eprints
 Pub Date:
 September 2016
 arXiv:
 arXiv:1609.06733
 Bibcode:
 2016arXiv160906733D
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 47A57;
 47B36 (Primary);
 30E05;
 30B70;
 42C05 (Secondary)
 EPrint:
 21 pages