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

Abstract

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 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 −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.

A generalized inverse eigenvalue problem and m-functions

“…Remark 3.5. The emphasis on b j /d j being purely imaginary arises from the particular form (1.4) of the pencil (zJ [0,n] −H [0,n] ), where in fact b j = id j and c j = 1, j = 0, 1, • • • , n. As mentioned in Section 1, such pencils appear in analytic function theory and a case has been made to call such pencils as Wall pencils [8].…”

“…that is, the b s appearing in H [0,n] are purely imaginary: b j = ib j , d j = b j and the c s appearing in J [0,n] are unity. The matrix pencil (1.4) has its origins in the continued fraction representation of Nevanlinna functions, which in turn are obtained via the Cayley transformation of the continued fraction representation of a Carathéodory function [14] (see also [8]). As further specific illustrations, the rational functions arising as components of eigenvectors in such cases have been related to a class of hypergeometric polynomials orthogonal on the unit circle [10] as well as pseudo-Jacobi polynomials (or Routh-Romanovski polynomials) [8].…”

“…is also of the form (1.4)? In the direct problem, Wall's algorithm [14, Chapter XV] (see also [8]) generates the sequences {c j , b j , d j } so that the above question is answered in the affirmative (in fact for H [0,∞] − λJ [0,∞] ). However, we will see that this is not so when we consider the inverse case.…”

A generalization of Schur functions: Applications to Nevanlinna functions, orthogonal polynomials, random walks and unitary and open quantum walks

“…The algorithm (56) is considered sometimes as the Schur algorithm for Nevanlinna functions of Stieltjes form, see [3, Chapter 3] and [5,31]. However, (54) seems a more natural version of the standard Schur algorithm (10).…”

“…Mappings between the open unit disk and the open upper half-plane may be useful for surmising results for Nevanlinna functions inspired by the case of Schur functions, but this is not the end of the story. For instance, these transformations do not suggest the simplicity of taking the point at infinity as interpolation point in a Schur algorithm for Nevanlinna functions (concerning the differences beween the interpolation problems for Schur and Nevanlinna functions, see [31]). Besides, such mappings have not been used to understand the Nevanlinna version of factorizations already known for Schur functions.…”