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

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