The functions 
generated by Schur's algorithm are related by
where 
denotes the elementary linear fractional transformation
(LFT)
We therefore have
, where 
is the composition
In this way we obtain the Schur continued fraction representation
of 
,
The function 
is referred to as the nth approximant
of 
, and 
is called the nth tail
of 
.
It is shown in [5], and easily verified, that
the LFT 
can be expressed as
where the polynomials 
and 
satisfy the recurrence relations
It follows by induction that
and 
have degree less than n,
and
,
,
, and 
for all n.
The polynomials 
and 
are generalizations of the
Schur parameters in the sense that they determine a composition of
the elementary linear fractional transformations determined by
the Schur parameters.
We will refer to 
and 
as the nth
Schur polynomials
associated with the Schur function 
.
The following result
shows how the Schur polynomials provide the Szego polynomial
, and hence provide
for the first phase of a Toeplitz solver.
The proof follows by comparison of the recurrence relations for
with those for 
and 
[5].
