  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.
