Orthogonal Rational Functions on the Unit Circle Orthogonal Rational Functions
on the Unit Circle
with Prescribed Poles not on the Unit Circle††This paper is a contribution to
the Special Issue on Orthogonal Polynomials, Special Functions and
Applications (OPSFA14). The full collection is available at
https://www.emis.de/journals/SIGMA/OPSFA2017.html

Adhemar BULTHEEL †, Ruyman CRUZ-BARROSO ‡ and Andreas LASAROW §

A. Bultheel, R. Cruz-Barroso and A. Lasarow

† Department of Computer Science, KU Leuven, Belgium
<EMAIL_ADDRESS>https://people.cs.kuleuven.be/~adhemar.bultheel/

‡ Department of Mathematical Analysis, La Laguna University, Tenerife, Spain
<EMAIL_ADDRESS>
§ Fak. Informatik, Mathematik & Naturwissenschaften, HTWK Leipzig, Germany
<EMAIL_ADDRESS>
Received August 01, 2017, in final form November 20, 2017; Published online
December 03, 2017

Orthogonal rational functions (ORF) on the unit circle generalize orthogonal
polynomials (poles at infinity) and Laurent polynomials (poles at zero and
infinity). In this paper we investigate the properties of and the relation
between these ORF when the poles are all outside or all inside the unit disk,
or when they can be anywhere in the extended complex plane outside the unit
circle. Some properties of matrices that are the product of elementary unitary
transformations will be proved and some connections with related algorithms
for direct and inverse eigenvalue problems will be explained.

orthogonal rational functions; rational Szegő quadrature; spectral method;
rational Krylov method; AMPD matrix 30D15; 30E05; 42C05; 44A60

##  1 Introduction

Orthogonal rational functions (ORF) on the unit circle are well known as
generalizations of orthogonal polynomials on the unit circle (OPUC). The pole
at infinity of the polynomials is replaced by poles “in the neighborhood” of
infinity, i.e., poles outside the closed unit disk. The recurrence relations
for the ORF generalize the Szegő recurrence relations for the polynomials.

If $\mu$ is the orthogonality measure supported on the unit circle, and
$L^{2}_{\mu}$ the corresponding Hilbert space, then the shift operator
$\mathcal{T}_{\mu}\colon L^{2}_{\mu}\to L^{2}_{\mu}\colon f(z)\mapsto zf(z)$
restricted to the polynomials has a representation with respect to the
orthogonal polynomials that is a Hessenberg matrix. However, if instead of a
polynomial basis, one uses a basis of orthogonal Laurent polynomials by
alternating between poles at infinity and poles at the origin, a full unitary
representation of $\mathcal{T}_{\mu}$ with respect to this basis is a five-
diagonal CMV matrix [12].

The previous ideas have been generalized to the rational case by Velázquez in
[47]. He showed that the representation of the shift operator with respect to
the classical ORF is not a Hessenberg matrix but a matrix Möbius transform of
a Hessenberg matrix. However, a full unitary representation can be obtained if
the shift is represented with respect to a rational analog of the Laurent
polynomials by alternating between a pole inside and a pole outside the unit
disk. The resulting matrix is a matrix Möbius transform of a five-diagonal
matrix.

Orthogonal Laurent polynomials on the real line, a half-line, or an interval
were introduced by Jones et al. [31, 32] in the context of moment problems,
Padé approximation and quadrature and this was elaborated by many authors.
González-Vera and his co-workers were in particular involved in extending the
theory where the poles zero and infinity alternate (the so-called balanced
situation) to a more general case where in each step either infinity or zero
can be chosen as a pole in any arbitrary order [8, 20]. They also identify the
resulting orthogonal Laurent polynomials as shifted versions of the orthogonal
polynomials. Hence the orthogonal Laurent polynomials satisfy the same
recurrence as the classical orthogonal polynomials after an appropriate
shifting and normalization is embedded.

The corresponding case of orthogonal Laurent polynomials on the unit circle
was introduced by Thron in [40] and has been studied more recently in for
example [15, 18]. Papers traditionally deal with the balanced situation like
in [18] but in [15] also an arbitrary ordering was considered. Only in [16]
Cruz-Barroso and Delvaux investigated the structure of the matrix
representation with respect to the basis of the resulting orthogonal Laurent
polynomials on the circle. They called it a “snake-shaped” matrix which
generalizes the five diagonal matrix.

The purpose of this paper is to generalize these ideas valid for Laurent
polynomials on the circle to the rational case. That is to choose the poles of
the ORF in an arbitrary order either inside or outside the unit disk. We
relate the resulting ORF with the ORF having all their poles outside or all
their poles inside the disk, and study the corresponding recurrence relations.
With respect to this new orthogonal rational basis, the shift operator will be
represented by a matrix Möbius transformation of a snake-shaped matrix.

In the papers by Lasarow and coworkers (e.g., [23, 24, 25, 34]) matrix
versions of the ORF are considered. In these papers also an arbitrary choice
of the poles is allowed but only with the restrictive condition that if
$\alpha$ is used as a pole, then $1/\overline{\alpha}$ cannot be used anymore.
This means that for example the “balanced situation” is excluded. One of the
goals of this paper is to remove this restriction on the poles.

In the context of quadrature formulas, an arbitrary sequence of poles not on
the unit circle was also briefly discussed in [19]. The sequence of poles
considered there need not be Newtonian, i.e., the poles for the ORF of degree
$n$ may depend on $n$. Since our approach will emphasize the role of the
recurrence relation for the ORF, we do need a Newtonian sequence, although
some of the results may be generalizable to the situation of a non-Newtonian
sequence of poles.

One of the applications of the theory of ORF is the construction of quadrature
formulas on the unit circle that form rational generalizations of the Szegő
quadrature. They are exact in spaces of rational functions having poles inside
and outside the unit disk. The nodes of the quadrature formula are zeros of
para-orthogonal rational functions (PORF) and the weights are all positive
numbers. These nodes and weights can (like in Gaussian quadrature) be derived
from the eigenvalue decomposition of a unitary truncation of the shift
operator to a finite-dimensional subspace. One of the results of the paper is
that there is no gain in considering an arbitrary sequence of poles inside and
outside the unit disk unless in a balanced situation. When all the poles are
chosen outside the closed unit disk or when some of them are reflected in the
circle, the same quadrature formula will be obtained. The computational effort
for the general case will not increase but neither can it reduce the cost.

In network applications or differential equations one often has to work with
functions of large sparse matrices. If $A$ is a matrix and the matrix function
$f(A)$ allows the Cauchy representation
$f(A)=\int_{\Gamma}f(z)(z-A)^{-1}\mathrm{d}\mu(z)$, where $\Gamma$ is a
contour encircling all the eigenvalues of $A$ then numerical quadrature is a
possible technique to obtain an approximation for $f(A)$. If for example
$\Gamma$ is the unit circle, then expressions like $u^{*}f(A)u$ for some
vector $u$ can be approximated by quadrature formulas discussed in this paper
which will be implemented disguised as Krylov subspace methods (see for
example [27, 29, 33]).

The purpose of the paper though is not to discuss quadrature in particular. It
is just an example application that does not require much extra introduction
of new terminology and notation. The main purpose however is to give a general
framework on which to build for the many applications of ORFs. Just like
orthogonal polynomials are used in about every branch of mathematics, ORFs can
be used with the extra freedom to exploit the location of the poles. For
example, it can be shown that the ORFs can be used to solve multipoint moment
problems as well as more general rational interpolation problems where
locations of the poles inside and outside the circle are important for the
engineering applications like system identification, model reduction,
filtering, etc. When modelling the transfer function of a linear system, poles
should be chosen inside as well as outside the disk to guarantee that the
transient as well as the steady state of the system is well modelled. It would
lead us too far to also include the interpolation properties of multipoint
Padé approximation and the related applications in several branches of
engineering. We only provide the basics in this paper so that it can be used
in the context of more applied papers.

The interpretation of the recursion for the ORFs as a factorization of a
matrix into elementary unitary transformations illustrates that the spectrum
of the resulting matrix is independent of the order in which the elementary
factors are multiplied. As far as we know, this fact was previously unknown in
the linear algebra community, unless in particular cases like unitary
Hessenberg matrices. As an illustration, we develop some preliminary results
in Section 11 in a linear algebra setting that is slightly more general than
the ORF situation.

In the last decades, many papers appeared on inverse eigenvalue problems for
unitary Hessenberg matrices and rational Krylov methods. Some examples are [4,
30, 35, 36, 37, 38, 44]. These use elementary operations that are very closely
related to the recurrence that will be discussed in this paper. However, they
are not the same and often miss the flexibility discussed here. We shall
illustrate some of these connections with certain algorithms from the
literature in Section 12.

The outline of the paper is as follows. In Section 2 we introduce the main
notations used in this paper. The linear spaces and the ORF bases are given in
Section 3. Section 4 brings the Christoffel–Darboux relations and the
reproducing kernels which form an essential element to obtain the recurrence
relation given in Section 5 but also for the PORF in Section 6 to be used for
quadrature formulas in Section 7. The alternative representation of the shift
operator is given in Section 8 and its factorization in elementary $2\times 2$
blocks in the subsequent Section 9. We end by drawing some conclusions about
the spectrum of the shift operator and about the computation of rational Szegő
quadrature formulas in Section 10. The ideas that we present in this paper,
especially the factorization of unitary Hessenberg matrices in elementary
unitary factors is also used in the linear algebra literature mostly in the
finite-dimensional situation. These elementary factors and what can be said
about the spectrum of their product is the subject of Section 11. These
elementary unitary transformations are intensively used in numerical
algorithms such as Arnoldi-based Krylov methods where they are known as core
transformations. Several variants of these rational Krylov methods exist. The
algorithms are quite similar yet different from our ORF recursion as we
explain briefly in Section 12 illustrating why we believe the version
presented in this paper has superior advantages.

##  2 Basic definitions and notation

We use the following notation. $\mathbb{C}$ denotes the complex plane,
$\widehat{\mathbb{C}}$ the extended complex plane (one point
compactification), $\mathbb{R}$ the real line, $\widehat{\mathbb{R}}$ the
closure of $\mathbb{R}$ in $\widehat{\mathbb{C}}$, $\mathbb{T}$ the unit
circle, $\mathbb{D}$ the open unit disk,
$\widehat{\mathbb{D}}=\mathbb{D}\cup\mathbb{T}$, and
$\mathbb{E}=\widehat{\mathbb{C}}\setminus\widehat{\mathbb{D}}$. For any number
$z\in\widehat{\mathbb{C}}$ we define $z_{*}=1/\overline{z}$ (and set
$1/0=\infty$, $1/\infty=0$) and for any complex function $f$, we define
$f_{*}(z)=\overline{f(z_{*})}$.

To approximate an integral

$\displaystyle I_{\mu}(f)=\int_{\mathbb{T}}f(z)\mathrm{d}\mu(z),$

where $\mu$ is a probability measure on $\mathbb{T}$ one may use Szegő
quadrature formulas. The nodes of this quadrature can be computed by using the
Szegő polynomials. Orthogonality in this paper will always be with respect to
the inner product

$\displaystyle\left\langle
f,g\right\rangle=\int_{\mathbb{T}}\overline{f(z)}g(z)\mathrm{d}\mu(z).$

The weights of the $n$-point quadrature are all positive, the nodes are on
$\mathbb{T}$ and the formula is exact for all Laurent polynomials
$f\in\mathop{\rm span}\nolimits\\{z^{k}\colon|k|\leq n-1\\}$.

This has been generalized to rational functions with a set of predefined
poles. The corresponding quadrature formulas are then rational Szegő
quadratures. This has been discussed in many papers and some of the earlier
results were summarized in the book [9]. We briefly recall some of the results
that are derived there. The idea is the following. Fix a sequence
$\underline{\mbox{\boldmath{$\alpha$}}}=(\alpha_{k})_{k\in\mathbb{N}}$ with
${\mbox{\boldmath{$\alpha$}}}=\\{\alpha_{k}\\}_{k\in\mathbb{N}}\subset\mathbb{D}$,
and consider the subspaces of rational functions defined by

$\displaystyle\mathcal{L}_{0}=\mathbb{C},\qquad\mathcal{L}_{n}=\left\\{\frac{p_{n}(z)}{\pi_{n}(z)}\colon
p_{n}\in\mathcal{P}_{n},\
\pi_{n}(z)=\prod_{k=1}^{n}(1-\overline{\alpha}_{k}z)\right\\},\qquad n\geq 1,$

where $\mathcal{P}_{n}$ is the set of polynomials of degree at most $n$. These
rational functions have their poles among the points in
${\mbox{\boldmath{$\alpha$}}}_{*}=\\{\alpha_{j*}=1/\overline{\alpha}_{j}\colon\alpha_{j}\in{\mbox{\boldmath{$\alpha$}}}\\}$.
We denote the corresponding sequence as
$\underline{\mbox{\boldmath{$\alpha$}}}_{*}=(\alpha_{j*})_{j\in\mathbb{N}}$.
Let $\phi_{n}\in\mathcal{L}_{n}\setminus\mathcal{L}_{n-1}$, and
$\phi_{n}\perp\mathcal{L}_{n-1}$ be the $n$th orthogonal rational basis
function (ORF) in a nested sequence. It is well known that these functions
have all their zeros in $\mathbb{D}$ (see, e.g., [9, Corollary 3.1.4]).
However, the quadrature formulas we have in mind should have their nodes on
the circle $\mathbb{T}$. Therefore, para-orthogonal rational functions (PORF)
are introduced. They are defined by

$\displaystyle
Q_{n}(z,\tau)=\phi_{n}(z)+\tau\phi_{n}^{*}(z),\qquad\tau\in\mathbb{T},$

where besides the ORF $\phi_{n}(z)=\frac{p_{n}(z)}{\pi_{n}(z)}$, also the
“reciprocal” function
$\phi_{n}^{*}(z)=\frac{p_{n}^{*}(z)}{\pi_{n}(z)}=\frac{z^{n}p_{n*}(z)}{\pi_{n}(z)}$
is introduced. These PORF have $n$ simple zeros
$\\{\xi_{nk}\\}_{k=1}^{n}\subset\mathbb{T}$ (see, e.g., [9, Theorem 5.2.1]) so
that they can be used as nodes for the quadrature formulas

$\displaystyle I_{n}(f)=\sum_{k=1}^{n}w_{nk}f(\xi_{nk})$

and the weights are all positive, given by
$w_{nk}=1/\sum\limits_{j=0}^{n-1}|\phi_{j}(\xi_{nj})|^{2}$ (see, e.g., [9,
Theorem 5.4.2]). These quadrature formulas are exact for all functions of the
form
$\\{f=g_{*}h\colon\\!g,h\in\mathcal{L}_{n-1}\\}=\mathcal{L}_{n-1}\mathcal{L}_{(n-1)*}$
(see, e.g., [9, Theorem 5.3.4]).

The purpose of this paper is to generalize the situation where the
$\alpha_{j}$ are all in $\mathbb{D}$ to the situation where they are anywhere
in the extended complex plane outside $\mathbb{T}$. This will require the
introduction of some new notation.

So consider a sequence $\underline{\mbox{\boldmath{$\alpha$}}}$ with
${\mbox{\boldmath{$\alpha$}}}\subset\mathbb{D}$ and its reflection in the
circle $\underline{\mbox{\boldmath{$\beta$}}}=(\beta_{j})_{j\in\mathbb{N}}$
where $\beta_{j}=1/\overline{\alpha}_{j}=\alpha_{j*}\in\mathbb{E}$. We now
construct a new sequence
$\underline{\mbox{\boldmath{$\gamma$}}}=(\gamma_{j})_{j\in\mathbb{N}}$ where
each $\gamma_{j}$ is either equal to $\alpha_{j}$ or $\beta_{j}$.

Partition $\\{1,2,\ldots,n\\}$
($n\in\widehat{\mathbb{N}}=\mathbb{N}\cup\\{\infty\\}$) into two disjoint
index sets: the ones where $\gamma_{j}=\alpha_{j}$ and the indices where
$\gamma_{j}=\beta_{j}$:

$\displaystyle\mathbbm{a}_{n}=\\{j\colon\gamma_{j}=\alpha_{j}\in\mathbb{D},~{}1\leq
j\leq
n\\}\qquad\mbox{and}\qquad\mathbbm{b}_{n}=\\{j\colon\gamma_{j}=\beta_{j}\in\mathbb{E},~{}1\leq
j\leq n\\},$

and define

$\displaystyle{\mbox{\boldmath{$\alpha$}}}_{n}=\\{\alpha_{j}\colon
j\in\mathbbm{a}_{n}\\}\qquad\mbox{and}\qquad{\mbox{\boldmath{$\beta$}}}_{n}=\\{\beta_{j}\colon
j\in\mathbbm{b}_{n}\\}.$

It will be useful to prepend the sequence
$\underline{\mbox{\boldmath{$\alpha$}}}$ with an extra point $\alpha_{0}=0$.
That means that $\underline{\mbox{\boldmath{$\beta$}}}$ is preceded by
$\beta_{0}=1/\overline{\alpha}_{0}=\infty$. For
$\underline{\mbox{\boldmath{$\gamma$}}}$, the initial point can be
$\gamma_{0}=\alpha_{0}=0$ or $\gamma_{0}=\beta_{0}=\infty$.

With each of the sequences $\underline{\mbox{\boldmath{$\alpha$}}}$,
$\underline{\mbox{\boldmath{$\beta$}}}$, and
$\underline{\mbox{\boldmath{$\gamma$}}}$ we can associate orthogonal rational
functions. They will be closely related as we shall show. The ORF for the
$\underline{\mbox{\boldmath{$\gamma$}}}$ sequence can be derived from the ORF
for the $\underline{\mbox{\boldmath{$\alpha$}}}$ sequence by multiplying with
a Blaschke product just like the orthogonal Laurent polynomials are
essentially shifted versions of the orthogonal polynomials (see, e.g., [15]).

To define the denominators of our rational functions, we introduce the
following elementary factors:

$\displaystyle\varpi_{j}^{\alpha}(z)=1-\overline{\alpha}_{j}z,\qquad\\!\varpi_{j}^{\beta}(z)=\begin{cases}1-\overline{\beta}_{j}z,&\text{if
}\beta_{j}\neq\infty,\\\ -z,&\text{if
}\beta_{j}=\infty,\end{cases}\qquad\\!\varpi_{j}^{\gamma}(z)=\begin{cases}\varpi_{j}^{\alpha}(z),&\text{if
}\gamma_{j}=\alpha_{n},\\\ \varpi_{j}^{\beta}(z),&\text{if
}\gamma_{j}=\beta_{n}.\end{cases}$

Note that if $\alpha_{j}=0$ and hence $\beta_{j}=\infty$ then
$\varpi_{j}^{\alpha}(z)=1$ but $\varpi_{j}^{\beta}(z)=-z$.

To separate the $\alpha$ and the $\beta$-factors in a product, we also define

$\displaystyle\dot{\varpi}_{j}^{\alpha}(z)=\begin{cases}\varpi_{j}^{\alpha},&\text{if
}\gamma_{j}=\alpha_{j},\\\ 1,&\text{if
}\gamma_{j}=\beta_{j},\end{cases}\qquad\mbox{and}\qquad\dot{\varpi}_{j}^{\beta}(z)=\begin{cases}\varpi_{j}^{\beta},&\text{if
}\gamma_{j}=\beta_{j},\\\ 1,&\text{if }\gamma_{j}=\alpha_{j}.\end{cases}$

Because the sequence $\underline{\mbox{\boldmath{$\gamma$}}}$ is our main
focus, we simplify the notation by removing the superscript $\gamma$ when not
needed, e.g.,
$\varpi_{j}=\varpi_{j}^{\gamma}=\dot{\varpi}_{j}^{\alpha}\dot{\varpi}_{j}^{\beta}$
etc.

We can now define for $\nu\in\\{\alpha,\beta,\gamma\\}$

$\displaystyle\pi_{n}^{\nu}(z)=\prod_{j=1}^{n}\varpi_{j}^{\nu}(z)$

and the reduced products separating the $\alpha$ and the $\beta$-factors

$\displaystyle\dot{\pi}_{n}^{\alpha}(z)=\prod_{j=1}^{n}\dot{\varpi}_{j}^{\alpha}(z)=\prod_{j\in\mathbbm{a}_{n}}\varpi_{j}(z),\qquad\dot{\pi}_{n}^{\beta}(z)=\prod_{j=1}^{n}\dot{\varpi}_{j}^{\beta}(z)=\prod_{j\in\mathbbm{b}_{n}}\varpi_{j}(z),$

so that

$\displaystyle\pi_{n}(z)=\prod_{j=1}^{n}\varpi_{j}(z)=\dot{\pi}_{n}^{\alpha}(z)\dot{\pi}_{n}^{\beta}(z).$

We assume here and in the rest of the paper that products over
$j\in\varnothing$ equal 1.

The Blaschke factors are defined for $\nu\in\\{\alpha,\beta,\gamma\\}$ as

$\displaystyle\zeta^{\nu}_{j}(z)=\sigma_{j}^{\nu}\displaystyle\frac{z-\nu_{j}}{1-\overline{\nu}_{j}z},\qquad$
$\displaystyle\sigma^{\nu}_{j}=\frac{\overline{\nu}_{j}}{|\nu_{j}|},\qquad$
$\displaystyle\text{if }\nu_{j}\not\in\\{0,\infty\\},$
$\displaystyle\zeta^{\nu}_{j}(z)=\sigma^{\nu}_{j}z=z,\qquad$
$\displaystyle\sigma^{\nu}_{j}=1,\qquad$ $\displaystyle\text{if }\nu_{j}=0,$
$\displaystyle\zeta^{\nu}_{j}(z)=\sigma^{\nu}_{j}/z=1/z,\qquad$
$\displaystyle\sigma^{\nu}_{j}=1,\qquad$ $\displaystyle\text{if
}\nu_{j}=\infty.$

Thus

$\displaystyle\sigma^{\nu}_{j}=\begin{cases}\dfrac{\overline{\nu}_{j}}{|\nu_{j}|},&\text{for}~{}~{}\nu_{j}\not\in\\{0,\infty\\},\\\
1,&\text{for}~{}~{}\nu_{j}\in\\{0,\infty\\}.\end{cases}$

Because $\sigma_{n}^{\alpha}=\sigma_{n}^{\beta}$, we can remove the
superscript and just write $\sigma_{n}$. If we also use the following notation
which maps complex numbers onto $\mathbb{T}$

$\displaystyle\mathfrak{u}(z)=\begin{cases}\dfrac{\overline{z}}{|z|}\in\mathbb{T},&z\in\mathbb{C}\setminus\\{0\\},\\\
1,&z\in\\{0,\infty\\},\end{cases}$

then
$\sigma_{j}=\mathfrak{u}(\alpha_{j})=\mathfrak{u}(\beta_{j})=\mathfrak{u}(\gamma_{j})$.

Set $(\varpi_{j}^{\nu})^{*}(z)=\varpi_{j}^{\nu*}(z)=z\varpi^{\nu}_{j*}(z)$
(e.g., $(1-\overline{\alpha}_{j}z)^{*}=z-\alpha_{j}$ if $\nu=\alpha$), then
$\zeta_{j}^{\nu}=\sigma_{j}\frac{\varpi_{j}^{\nu*}}{\varpi_{j}^{\nu}}$. Later
we shall also use $\pi_{n}^{\nu*}$ to mean
$\prod\limits_{j=1}^{n}\varpi_{j}^{\nu*}$. Note that
$\zeta_{j}^{\alpha}=\zeta_{j*}^{\beta}=1/\zeta_{j}^{\beta}$. Moreover if
$\alpha_{j}=0$ and hence $\beta_{j}=\infty$, then $\varpi_{j}^{\alpha*}(z)=z$
and $\varpi_{j}^{\beta*}(z)=-1$.

Next define the finite Blaschke products for $\nu\in\\{\alpha,\beta\\}$

$\displaystyle B_{0}^{\nu}=1,\qquad\mbox{and}\qquad
B_{n}^{\nu}(z)=\prod_{j=1}^{n}\zeta_{j}^{\nu}(z),\qquad n=1,2,\ldots.$

It is important to note that here $\nu\neq\gamma$. For the definition of
$B_{n}^{\gamma}=B_{n}$ see below.

Like we have split up the denominators
$\pi_{n}=\dot{\pi}_{n}^{\alpha}\dot{\pi}_{n}^{\beta}$ in the $\alpha$-factors
and the $\beta$-factors, we also define for $n\geq 1$

$\displaystyle\dot{\zeta}_{j}^{\alpha}=\begin{cases}\zeta_{j}^{\alpha},&\mbox{if~{}~{}}\gamma_{j}=\alpha_{j},\\\
1,&\mbox{if~{}~{}}\gamma_{j}=\beta_{j},\end{cases}\qquad\dot{\zeta}_{j}^{\beta}=\begin{cases}\zeta_{j}^{\beta},&\mbox{if~{}~{}}\gamma_{j}=\beta_{j},\\\
1,&\mbox{if~{}~{}}\gamma_{j}=\alpha_{j},\end{cases}$

and

$\displaystyle\dot{B}_{n}^{\alpha}(z)=\prod_{j=1}^{n}\dot{\zeta}_{j}^{\alpha}(z)=\prod_{j\in\mathbbm{a}_{n}}\zeta_{j}(z),\qquad\mbox{and}\qquad\dot{B}_{n}^{\beta}(z)=\prod_{j=1}^{n}\dot{\zeta}_{j}^{\beta}(z)=\prod_{j\in\mathbbm{b}_{n}}\zeta_{j}(z),$

so that we can define the finite Blaschke products for the
$\underline{\mbox{\boldmath{$\gamma$}}}$ sequence:

$\displaystyle
B_{n}(z)=\begin{cases}\dot{B}_{n}^{\alpha}(z),&\mbox{if~{}~{}}\gamma_{n}=\alpha_{n},\\\
\dot{B}_{n}^{\beta}(z),&\mbox{if~{}~{}}\gamma_{n}=\beta_{n}.\end{cases}$

Note that the reflection property of the factors also holds for the products:
$B_{n}^{\alpha}=(B_{n}^{\beta})_{*}=1/B_{n}^{\beta}$, $B_{n*}=1/B_{n}$, and
$(\dot{B}_{n}^{\alpha}\dot{B}_{n}^{\beta})_{*}=1/(\dot{B}_{n}^{\beta}\dot{B}_{n}^{\alpha})$.
However,

$\displaystyle\dot{B}_{n}^{\alpha}=\prod\limits_{j\in\mathbbm{a}_{n}}\zeta_{j}^{\alpha}=\prod\limits_{j\in\mathbbm{a}_{n}}\zeta_{j*}^{\beta}=\prod\limits_{j\in\mathbbm{a}_{n}}1/\zeta_{j}^{\beta}\neq\prod\limits_{j\in\mathbbm{b}_{n}}1/\zeta_{j}^{\beta}=1/\dot{B}_{n}^{\beta}.$

##  3 Linear spaces and ORF bases

We can now introduce our spaces of rational functions for $n\geq 0$:

$\displaystyle\mathcal{L}_{n}^{\nu}=\mathop{\rm
span}\nolimits\\{B_{0}^{\nu},B_{1}^{\nu},\ldots,B_{n}^{\nu}\\},\qquad\nu\in\\{\alpha,\beta,\gamma\\},\qquad\mbox{and}$
$\displaystyle\dot{\mathcal{L}}_{n}^{\nu}=\mathop{\rm
span}\nolimits\\{\dot{B}_{0}^{\nu},\dot{B}_{1}^{\nu},\ldots,\dot{B}_{n}^{\nu}\\},\qquad\nu\in\\{\alpha,\beta\\}.$

The dimension of $\mathcal{L}_{n}^{\nu}$ is $n+1$ for
$\nu\in\\{\alpha,\beta,\gamma\\}$, but note that the dimension of
$\dot{\mathcal{L}}_{n}^{\nu}$ for $\nu\in\\{\alpha,\beta\\}$ can be less than
$n+1$. Indeed some of the $\dot{B}_{j}^{\nu}$ may be repeated so that for
example the dimension of $\dot{\mathcal{L}}_{n}^{\alpha}$ is only
$|\mathbbm{a}_{n}|+1$ with $|\mathbbm{a}_{n}|$ the cardinality of
$\mathbbm{a}_{n}$ and similarly for $\nu=\beta$. Hence for $\nu=\gamma$:

$\displaystyle\mathcal{L}_{n}=\mathop{\rm
span}\nolimits\\{B_{0},\ldots,B_{n}\\}=\mathop{\rm
span}\nolimits\big{\\{}\dot{B}_{0},\dot{B}_{1}^{\alpha},\ldots,\dot{B}_{n}^{\alpha},\dot{B}_{1}^{\beta},\ldots,\dot{B}_{n}^{\beta}\big{\\}}=\dot{\mathcal{L}}_{n}^{\alpha}+\dot{\mathcal{L}}_{n}^{\beta}=\dot{\mathcal{L}}_{n}^{\alpha}\dot{\mathcal{L}}_{n}^{\beta}.$

Because for $n\geq 1$

$\displaystyle\dot{B}_{n}^{\alpha}=\prod_{j\in\mathbbm{a}_{n}}\zeta_{j}^{\alpha}=\prod_{j\in\mathbbm{a}_{n}}\frac{1}{\zeta_{j}^{\beta}}\qquad\mbox{and}\qquad\dot{B}_{n}^{\beta}=\prod_{j\in\mathbbm{b}_{n}}\zeta_{j}^{\beta}=\prod_{j\in\mathbbm{b}_{n}}\frac{1}{\zeta_{j}^{\alpha}},$

it should be clear that
$B_{k}^{\alpha}=\dot{B}_{k}^{\alpha}/\dot{B}_{k}^{\beta}$ and
$B_{k}^{\beta}=\dot{B}_{k}^{\beta}/\dot{B}_{k}^{\alpha}$, hence that

$\displaystyle\mathcal{L}_{n}^{\alpha}=\mathop{\rm
span}\nolimits\left\\{\dot{B}_{0},\frac{\dot{B}_{1}^{\alpha}}{\dot{B}_{1}^{\beta}},\ldots,\frac{\dot{B}_{n}^{\alpha}}{\dot{B}_{n}^{\beta}}\right\\}\qquad\mbox{and}\qquad\mathcal{L}_{n}^{\beta}=\mathop{\rm
span}\nolimits\left\\{\dot{B}_{0},\frac{\dot{B}_{1}^{\beta}}{\dot{B}_{1}^{\alpha}},\ldots,\frac{\dot{B}_{n}^{\beta}}{\dot{B}_{n}^{\alpha}}\right\\}.$

Occasionally we shall also need the notation

$\displaystyle\dot{\varsigma}_{n}^{\alpha}=\prod_{j\in\mathbbm{a}_{n}}\sigma_{j}\in\mathbb{T},\qquad\dot{\varsigma}_{n}^{\beta}=\prod_{j\in\mathbbm{b}_{n}}\sigma_{j}\in\mathbb{T},\qquad\mbox{and}\qquad\varsigma_{n}=\prod_{j=1}^{n}\sigma_{j}\in\mathbb{T}.$

######  Lemma 3.1.

If $f\in\mathcal{L}_{n}$ then
$f/\dot{B}_{n}^{\beta}\in\mathcal{L}_{n}^{\alpha}$ and
$f/\dot{B}_{n}^{\alpha}\in\mathcal{L}_{n}^{\beta}$. In other words
$\mathcal{L}_{n}=\dot{B}_{n}^{\beta}\mathcal{L}_{n}^{\alpha}=\dot{B}_{n}^{\alpha}\mathcal{L}_{n}^{\beta}$.
This is true for all $n\geq 0$ if we set
$\dot{B}_{0}^{\alpha}=\dot{B}_{0}^{\beta}=1$.

###### Proof.

This is trivial for $n=0$ since then $\mathcal{L}_{n}=\mathbb{C}$. If
$f\in\mathcal{L}_{n}$, and $n\geq 1$ then it is of the form

$\displaystyle
f(z)=\frac{p_{n}(z)}{\pi_{n}(z)}=\frac{p_{n}(z)}{\dot{\pi}_{n}^{\alpha}(z)\dot{\pi}_{n}^{\beta}(z)},\qquad
p_{n}\in\mathcal{P}_{n}.$

Therefore

$\displaystyle\frac{f(z)}{\dot{B}_{n}^{\beta}(z)}=\overline{\dot{\varsigma}}_{n}^{\beta}\frac{p_{n}(z)\dot{\pi}_{n}^{\beta}(z)}{\dot{\pi}_{n}^{\alpha}(z)\dot{\pi}_{n}^{\beta}(z)\dot{\pi}_{n}^{\beta*}(z)}=\overline{\dot{\varsigma}}_{n}^{\beta}\frac{p_{n}(z)}{\dot{\pi}_{n}^{\alpha}(z)\dot{\pi}_{n}^{\beta*}(z)}.$

Recall that $\varpi_{j}^{\beta*}=-1$ and $\sigma_{j}=1$ if $\beta_{j}=\infty$
(and hence $\alpha_{j}=0$), we can leave these factors out and we shall write
$\cdot$$\mathop{\prod}$ for the product instead of $\prod$, the dot meaning
that we leave out all the factors for which
$\alpha_{j}=1/\overline{\beta}_{j}=0$.

$\displaystyle\frac{\overline{\dot{\varsigma}}_{n}^{\beta}}{\dot{\pi}_{n}^{\beta*}(z)}=\mathop{{\hbox
to0.0pt{\hbox
to9.44447pt{\hss$\cdot$\hss}\hss}\hbox{$\displaystyle\mathop{\prod}$}}}_{j\in\mathbbm{b}_{n}}\frac{\beta_{j}}{|\beta_{j}|(z-\beta_{j})}=\mathop{{\hbox
to0.0pt{\hbox
to9.44447pt{\hss$\cdot$\hss}\hss}\hbox{$\displaystyle\mathop{\prod}$}}}_{j\in\mathbbm{b}_{n}}\frac{|\alpha_{j}|}{\overline{\alpha}_{j}(z-1/\overline{\alpha}_{j})}=\mathop{{\hbox
to0.0pt{\hbox
to9.44447pt{\hss$\cdot$\hss}\hss}\hbox{$\displaystyle\mathop{\prod}$}}}_{j\in\mathbbm{b}_{n}}\frac{-|\alpha_{j}|}{1-\overline{\alpha}_{j}z},$

and thus

$\displaystyle\frac{f(z)}{\dot{B}_{n}^{\beta}(z)}=c_{n}\frac{p_{n}(z)}{\prod\limits_{j=1}^{n}(1-\overline{\alpha}_{j}z)}\in\mathcal{L}_{n}^{\alpha},\qquad
c_{n}=\mathop{{\hbox to0.0pt{\hbox
to9.44447pt{\hss$\cdot$\hss}\hss}\hbox{$\displaystyle\mathop{\prod}$}}}_{j\in\mathbbm{b}_{n}}(-|\alpha_{j}|)\neq
0.$

The second part is similar. ∎

######  Lemma 3.2.

With our previous definitions we have for $n\geq 1$

$\displaystyle\dot{B}_{n}^{\beta}\mathcal{L}_{n-1}^{\alpha}=\mathop{\rm
span}\nolimits\left\\{B_{k}^{\alpha}\dot{B}_{n}^{\beta}=\frac{\dot{B}_{k}^{\alpha}}{\dot{B}_{k}^{\beta}}\dot{B}_{n}^{\beta}\colon
k=0,\ldots,n-1\right\\}$
$\displaystyle\hphantom{\dot{B}_{n}^{\beta}\mathcal{L}_{n-1}^{\alpha}}=\dot{\zeta}_{n}^{\beta}\mathop{\rm
span}\nolimits\\{B_{0},B_{1},\ldots,B_{n-1}\\}=\dot{\zeta}_{n}^{\beta}\mathcal{L}_{n-1},$

and similarly

$\displaystyle\dot{B}_{n}^{\alpha}\mathcal{L}_{n-1}^{\beta}=\mathop{\rm
span}\nolimits\left\\{B_{k}^{\beta}\dot{B}_{n}^{\alpha}=\frac{\dot{B}_{k}^{\beta}}{\dot{B}_{k}^{\alpha}}\dot{B}_{n}^{\alpha}\colon
k=0,\ldots,n-1\right\\}$
$\displaystyle\hphantom{\dot{B}_{n}^{\alpha}\mathcal{L}_{n-1}^{\beta}}{}=\dot{\zeta}_{n}^{\alpha}\mathop{\rm
span}\nolimits\\{B_{0},B_{1},\ldots,B_{n-1}\\}=\dot{\zeta}_{n}^{\alpha}\mathcal{L}_{n-1}.$

###### Proof.

By our previous lemma
$\dot{B}_{n}^{\beta}\mathcal{L}_{n-1}^{\alpha}=\dot{\zeta}_{n}^{\beta}\dot{B}_{n-1}^{\beta}\mathcal{L}_{n-1}^{\alpha}=\dot{\zeta}_{n}^{\beta}\mathcal{L}_{n-1}$.
The second relation is proved in a similar way. ∎

To introduce the sequences of orthogonal rational functions (ORF) for the
different sequences $\underline{\mbox{\boldmath{$\nu$}}}$,
$\nu\in\\{\alpha,\beta,\gamma\\}$ recall the inner product that we can write
with our $(\ )_{*}$-notation as $\left\langle
f,g\right\rangle=\int_{\mathbb{T}}f_{*}(z)g(z)\mathrm{d}\mu(z)$ where $\mu$ is
assumed to be a probability measure positive a.e. on $\mathbb{T}$.

Then the orthogonal rational functions (ORF) with respect to the sequence
$\underline{\mbox{\boldmath{$\nu$}}}$ with $\nu\in\\{\alpha,\beta,\gamma\\}$
are defined by
$\phi_{n}^{\nu}\in\mathcal{L}_{n}^{\nu}\setminus\mathcal{L}_{n-1}^{\nu}$ with
$\phi_{n}^{\nu}\perp\mathcal{L}_{n-1}^{\nu}$ for $n\geq 1$ and we choose
$\phi_{0}^{\nu}=1$.

######  Lemma 3.3.

The function $\phi_{n}^{\alpha}\dot{B}_{n}^{\beta}$ belongs to
$\mathcal{L}_{n}$ and it is orthogonal to the $n$-dimensional subspace
$\dot{\zeta}_{n}^{\beta}\mathcal{L}_{n-1}$ for all $n\geq 1$.

Similarly, the function $\phi_{n}^{\beta}\dot{B}_{n}^{\alpha}$ belongs to
$\mathcal{L}_{n}$ and it is orthogonal to the $n$-dimensional subspace
$\dot{\zeta}_{n}^{\alpha}\mathcal{L}_{n-1}$, $n\geq 1$.

###### Proof.

First note that $\phi_{n}^{\alpha}\dot{B}_{n}^{\beta}\in\mathcal{L}_{n}$ by
Lemma 3.1.

By definition $\phi_{n}^{\alpha}\perp\mathcal{L}_{n-1}^{\alpha}$. Thus by
Lemma 3.2 and because $\left\langle
f,g\right\rangle=\left\langle\dot{B}_{n}^{\nu}f,\dot{B}_{n}^{\nu}g\right\rangle$,

$\displaystyle\dot{B}_{n}^{\beta}\phi_{n}^{\alpha}\perp\dot{B}_{n}^{\beta}\mathcal{L}_{n-1}^{\alpha}=\dot{\zeta}_{n}^{\beta}\mathcal{L}_{n-1}.$

The second claim follows by symmetry. ∎

Note that $\dot{\zeta}_{n}^{\beta}\mathcal{L}_{n-1}=\mathcal{L}_{n-1}$ if
$\gamma_{n}=\alpha_{n}$. Thus, up to normalization,
$\phi_{n}^{\alpha}\dot{B}_{n}^{\beta}$ is the same as $\phi_{n}$ and
similarly, if $\gamma_{n}=\beta_{n}$ then $\phi_{n}$ and
$\phi_{n}^{\beta}\dot{B}_{n}^{\alpha}$ are the same up to normalization.

######  Lemma 3.4.

For $n\geq 1$ the function $\dot{B}_{n}^{\alpha}(\phi_{n}^{\alpha})_{*}$
belongs to $\mathcal{L}_{n}$ and it is orthogonal to
$\dot{\zeta}_{n}^{\alpha}\mathcal{L}_{n-1}$.

Similarly, for $n\geq 1$ the function
$\dot{B}_{n}^{\beta}(\phi_{n}^{\beta})_{*}$ belongs to $\mathcal{L}_{n}$ and
it is orthogonal to $\dot{\zeta}_{n}^{\beta}\mathcal{L}_{n-1}$.

###### Proof.

Since
$\phi_{n}^{\alpha}\dot{B}_{n}^{\beta}\perp\dot{\zeta}_{n}^{\beta}\mathcal{L}_{n-1}$,

$\displaystyle(\phi_{n}^{\alpha}\dot{B}_{n}^{\beta})_{*}\perp\dot{\zeta}_{n*}^{\beta}\mathcal{L}_{(n-1)*},$

and thus by Lemma 3.2 and because

$\displaystyle\dot{B}_{n-1}^{\alpha}\dot{B}_{n-1}^{\beta}\mathcal{L}_{(n-1)*}=\dot{B}_{n-1}^{\alpha}\dot{B}_{n-1}^{\beta}\frac{\mathcal{P}_{(n-1)*}}{\dot{\pi}_{(n-1)*}^{\alpha}\dot{\pi}_{(n-1)*}^{\beta}}=\frac{\mathcal{P}_{n-1}}{\dot{\pi}_{n-1}^{\alpha}\dot{\pi}_{n-1}^{\beta}}=\mathcal{L}_{n-1}$

it follows that

$\displaystyle\dot{B}_{n}^{\alpha}\phi_{n*}^{\alpha}=\dot{B}_{n}^{\alpha}\dot{B}_{n}^{\beta}(\phi_{n}^{\alpha}\dot{B}_{n}^{\beta})_{*}\perp\dot{\zeta}_{n}^{\alpha}\dot{B}_{n-1}^{\alpha}\dot{B}_{n-1}^{\beta}\mathcal{L}_{(n-1)*}=\dot{\zeta}_{n}^{\alpha}\mathcal{L}_{n-1}.$

The other claim follows by symmetry. ∎

We now define the reciprocal ORFs by (recall
$f_{*}(z)=\overline{f(1/\overline{z})}$)

$\displaystyle(\phi_{n}^{\nu})^{*}=B_{n}^{\nu}(\phi_{n}^{\nu})_{*},\qquad\nu\in\\{\alpha,\beta\\}.$

For the ORF in $\mathcal{L}_{n}$ however we set

$\displaystyle\phi_{n}^{*}=\dot{B}_{n}^{\alpha}\dot{B}_{n}^{\beta}(\phi_{n})_{*}.$

Note that by definition $B_{n}$ is either $\dot{B}_{n}^{\alpha}$ or
$\dot{B}_{n}^{\beta}$ depending on $\gamma_{n}$ being $\alpha_{n}$ or
$\beta_{n}$, while in the previous definition we do not multiply with $B_{n}$
but with the product $\dot{B}_{n}^{\alpha}\dot{B}_{n}^{\beta}$. The reason is
that we want the operation $(\ )^{*}$ to be a map from $\mathcal{L}_{n}^{\nu}$
to $\mathcal{L}_{n}^{\nu}$ for all $\nu\in\\{\alpha,\beta,\gamma\\}$.

######  Remark 3.5.

As the operation $(\ )^{*}$ is a map from $\mathcal{L}_{n}^{\nu}$ to
$\mathcal{L}_{n}^{\nu}$, it depends on $n$ and on $\nu$. So to make the
notation unambiguous we should in fact use something like $f^{[\nu,n]}$ if
$f\in\mathcal{L}_{n}^{\nu}$. However, in order not to overload our notation,
we shall stick to the notation $f^{*}$ since it should always be clear from
the context what the space is to which $f$ will belong. Note that we also used
the same notation to transform polynomials. This is just a special case of the
general definition. Indeed, a polynomial of degree $n$ belongs to
$\mathcal{L}_{n}^{\alpha}$ for a sequence
$\underline{\mbox{\boldmath{$\alpha$}}}$ where all $\alpha_{j}=0$,
$j=0,1,2,\ldots$ and for this sequence $B_{n}^{\alpha}(z)=z^{n}$.

Note that for a constant $a\in\mathcal{L}_{0}=\widehat{\mathbb{C}}$ we have
$a^{*}=\overline{a}$. Although $(\ )^{*}$ is mostly used for scalar
expressions, we shall occasionally use $A^{*}$ where $A$ is a matrix whose
elements are all in $\mathcal{L}_{n}$. Then the meaning is that we take the
$(\ )^{*}$ conjugate of each element in its transpose. Thus if $A$ is a
constant matrix, then $A^{*}$ has the usual meaning of the adjoint or complex
conjugate transpose of the matrix. We shall need this in Section 8.

######  Remark 3.6.

It might also be helpful for the further computations to note the following.
If $p_{n}$ is a polynomial of degree $n$ with a zero at $\xi$, then
$p_{n}^{*}$ will have a zero at $\xi_{*}=1/\overline{\xi}$. Hence, if
$\nu\in\\{\alpha,\beta,\gamma\\}$ and
$\phi_{n}^{\nu}=\frac{p_{n}^{\nu}}{\pi_{n}^{\nu}}$, then
$\phi_{n*}^{\nu}=\frac{p_{n*}^{\nu}}{\pi_{n*}^{\nu}}=\frac{p_{n}^{\nu*}}{\pi_{n}^{\nu*}}$.
We know by [9, Corollary 3.1.4] that $\phi_{n}^{\alpha}$ has all its zeros in
$\mathbb{D}$, hence $p_{n}^{\alpha}$ does not vanish in $\mathbb{E}$ and
$p_{n}^{\alpha*}$ does not vanish in $\mathbb{D}$. By symmetry
$\phi_{n}^{\beta}$ has all its zeros in $\mathbb{E}$ and $p_{n}^{\beta*}$ does
not vanish in $\mathbb{E}$. For the general $\phi_{n}$, it depends on
$\gamma_{n}$ being $\alpha_{n}$ or $\beta_{n}$. However from the relations
between $\phi_{n}$ and $\phi_{n}^{\alpha}$ or $\phi_{n}^{\beta}$ that will be
derived below, we will be able to guarantee that at least for $z=\nu_{n}$ we
have $\phi_{n}^{\nu*}(\nu_{n})\neq 0$ and $p_{n}^{\nu*}(\nu_{n})\neq 0$ for
all $\nu\in\\{\alpha,\beta,\gamma\\}$ (see Corollary 3.11 below).

The orthogonality conditions define $\phi_{n}$ and $\phi_{n}^{*}$ uniquely up
to normalization. So let us now make the ORFs unique by imposing an
appropriate normalization. First assume that from now on the $\phi_{n}^{\nu}$
refer to orthonormal functions in the sense that $\|\phi_{n}^{\nu}\|=1$. This
makes them unique up to a unimodular constant. Defining this constant is what
we shall do now.

Suppose $\gamma_{n}=\alpha_{n}$, then $\phi_{n}$ and
$\phi_{n}^{\alpha}\dot{B}_{n}^{\beta}$ are both in $\mathcal{L}_{n}$ and
orthogonal to $\mathcal{L}_{n-1}$ (Lemma 3.3). If we assume $\|\phi_{n}\|=1$
and $\|\phi_{n}^{\alpha}\|=1$, hence
$\|\phi_{n}^{\alpha}\dot{B}_{n}^{\beta}\|=\|\phi_{n}^{\alpha}\|=1$, it follows
that there must be some unimodular constant $s_{n}^{\alpha}\in\mathbb{T}$ such
that $\phi_{n}=s_{n}^{\alpha}\phi_{n}^{\alpha}\dot{B}_{n}^{\beta}$. Of course,
we have by symmetry that for $\gamma_{n}=\beta_{n}$, there is some
$s_{n}^{\beta}\in\mathbb{T}$ such that
$\phi_{n}=s_{n}^{\beta}\phi_{n}^{\beta}\dot{B}_{n}^{\alpha}$.

To define the unimodular factors $s_{n}^{\alpha}$ and $s_{n}^{\beta}$, we
first fix $\phi_{n}^{\alpha}$ and $\phi_{n}^{\beta}$ uniquely as follows.

We know that $\phi_{n}^{\alpha}$ has all its zeros in $\mathbb{D}$ and hence
$\phi_{n}^{\alpha*}$ has all its zeros in $\mathbb{E}$ so that
$\phi_{n}^{\alpha*}({\alpha}_{n})\neq 0$. Thus we can take
$\phi_{n}^{\alpha*}({\alpha}_{n})>0$ as a normalization for
$\phi_{n}^{\alpha}$. Similarly for $\phi_{n}^{\beta}$ we can normalize by
$\phi_{n}^{\beta*}({\beta}_{n})>0$. In both cases, we have made the leading
coefficient with respect to the basis $\\{B_{j}^{\nu}\\}_{j=0}^{n}$ positive
since
$\phi_{n}^{\alpha}(z)=\overline{\phi_{n}^{\alpha*}(\alpha_{n})}B_{n}^{\alpha}(z)+\psi_{n-1}^{\alpha}(z)$
with $\psi_{n-1}^{\alpha}\in\mathcal{L}_{n-1}^{\alpha}$ and
$\phi_{n}^{\beta}(z)=\overline{\phi_{n}^{\beta*}(\beta_{n})}B_{n}^{\beta}(z)+\psi_{n-1}^{\beta}(z)$
with $\psi_{n-1}^{\beta}\in\mathcal{L}_{n-1}^{\beta}$. Before we define the
normalization for the $\underline{\mbox{\boldmath{$\gamma$}}}$ sequence, we
prove the following lemma which is a consequence of the normalization of the
$\phi_{n}^{\alpha}$ and the $\phi_{n}^{\beta}$.

######  Lemma 3.7.

For the orthonormal ORFs, it holds that $\phi_{n}^{\alpha}=\phi_{n*}^{\beta}$
and
$(\phi_{n}^{\alpha})^{*}\dot{B}_{n}^{\beta}=\phi_{n}^{\beta}\dot{B}_{n}^{\alpha}$
and hence also
$(\phi_{n}^{\beta})^{*}\dot{B}_{n}^{\alpha}=\phi_{n}^{\alpha}\dot{B}_{n}^{\beta}$
for all $n\geq 0$.

###### Proof.

For $n=0$, this is trivial since
$\phi_{0},\phi_{0}^{\alpha},\phi_{0}^{\beta},\dot{B}_{0}^{\alpha}$ and
$\dot{B}_{0}^{\beta}$ are all equal to 1.

We give the proof for $n\geq 1$ and $\gamma_{n}=\alpha_{n}$ (for
$\gamma_{n}=\beta_{n}$, the proof is similar). Since by previous lemmas
$\dot{B}_{n}^{\beta}(\phi_{n}^{\beta})_{*}$ and
$\phi_{n}^{\alpha}\dot{B}_{n}^{\beta}$ are both in $\mathcal{L}_{n}$ and
orthogonal to $\mathcal{L}_{n-1}$, and since
$\|\dot{B}_{n}^{\beta}(\phi_{n}^{\beta})_{*}\|=\|\phi_{n}^{\beta}\|=1$ and
$\|\phi_{n}^{\alpha}\dot{B}_{n}^{\beta}\|=\|\phi_{n}^{\alpha}\|=1$, there must
be some $s_{n}\in\mathbb{T}$ such that

$\displaystyle
s_{n}\phi_{n}^{\alpha}\dot{B}_{n}^{\beta}=\phi_{n*}^{\beta}\dot{B}_{n}^{\beta}\qquad\text{or}\qquad
s_{n}\phi_{n}^{\alpha}=\phi_{n*}^{\beta}.$

Multiply with $B_{n}^{\beta}=B_{n*}^{\alpha}$ and evaluate at ${\beta}_{n}$ to
get
$s_{n}\phi_{n}^{\alpha}({\beta}_{n})B_{n*}^{\alpha}({\beta}_{n})=\phi_{n}^{\beta*}({\beta}_{n})>0$.
Thus $s_{n}$ should arrange for

$\displaystyle
0<s_{n}\phi_{n}^{\alpha}(1/\overline{{\alpha}}_{n})B_{n*}^{\alpha}(1/\overline{{\alpha}}_{n})=s_{n}\overline{\phi_{n*}^{\alpha}({\alpha}_{n})}\overline{B_{n}^{\alpha}({\alpha}_{n})}=s_{n}\overline{\phi_{n}^{\alpha*}({\alpha}_{n})},$

and since $\phi_{n}^{\alpha*}({\alpha}_{n})>0$, it follows that $s_{n}=1$.

Because
$(\phi_{n}^{\alpha})^{*}=B_{n}^{\alpha}\phi_{n*}^{\alpha}=B_{n}^{\alpha}\phi_{n}^{\beta}$
and $B_{n}^{\alpha}=\dot{B}_{n}^{\alpha}/\dot{B}_{n}^{\beta}$, also the other
claims follow. ∎

For the normalization of the $\phi_{n}$, we can do two things: either we make
the normalization of $\phi_{n}$ simple and choose for example
$\phi_{n}^{*}({\gamma}_{n})>0$, similar to what we did for $\phi_{n}^{\alpha}$
and $\phi_{n}^{\beta}$ (but this is somewhat problematic as we shall see
below), or we can insist on keeping the relation with $\phi_{n}^{\alpha}$ and
$\phi_{n}^{\beta}$ simple as in the previous lemma and arrange that
$s_{n}^{\alpha}=s_{n}^{\beta}=1$. We choose for the second option.

Let us assume that $\gamma_{n}=\alpha_{n}$. Denote

$\displaystyle\phi_{n}(z)=\frac{p_{n}(z)}{\dot{\pi}_{n}^{\alpha}(z)\dot{\pi}_{n}^{\beta}(z)}\qquad\text{and}\qquad\phi_{n}^{\alpha}(z)=\frac{p_{n}^{\alpha}(z)}{\pi_{n}^{\alpha}(z)},$

with $p_{n}$ and $p_{n}^{\alpha}$ both polynomials in $\mathcal{P}_{n}$. Then

$\displaystyle\phi_{n}^{*}(z)=\frac{\varsigma_{n}\,p_{n}^{*}(z)}{\dot{\pi}_{n}^{\alpha}(z)\dot{\pi}_{n}^{\beta}(z)}\qquad\mbox{and}\qquad\phi_{n}^{\alpha*}(z)=\frac{\varsigma_{n}\,p_{n}^{\alpha*}(z)}{\pi_{n}^{\alpha}(z)},\qquad\varsigma_{n}=\prod_{j=1}^{n}\sigma_{j}.$

We already know that there is some $s_{n}^{\alpha}\in\mathbb{T}$ such that
$\phi_{n}=s_{n}^{\alpha}\dot{B}_{n}^{\beta}\phi_{n}^{\alpha}$. Take the $(\
)_{*}$ conjugate and multiply with $\dot{B}_{n}^{\alpha}\dot{B}_{n}^{\beta}$
to get
$\phi_{n}^{*}=\overline{s_{n}^{\alpha}}\dot{B}_{n}^{\beta}\phi_{n}^{\alpha*}$.

It now takes some simple algebra to reformulate
$\phi_{n}^{*}=\overline{s_{n}^{\alpha}}\dot{B}_{n}^{\beta}\phi_{n}^{\alpha*}$
as

$\displaystyle\phi_{n}^{*}(z)=\frac{\varsigma_{n}\,p_{n}^{*}(z)}{\dot{\pi}_{n}^{\alpha}(z)\dot{\pi}_{n}^{\beta}(z)}=\overline{s_{n}^{\alpha}}\frac{\varsigma_{n}\,p_{n}^{\alpha*}(z)}{\dot{\pi}_{n}^{\alpha}(z)\dot{\pi}_{n}^{\beta}(z)}\mathop{{\hbox
to0.0pt{\hbox
to9.44447pt{\hss$\cdot$\hss}\hss}\hbox{$\displaystyle\mathop{\prod}$}}}_{j\in\mathbbm{b}_{n}}(-|\beta_{j}|).$

This implies that
$p_{n}^{*}(z)=\overline{s_{n}^{\alpha}}p_{n}^{\alpha*}(z){\hbox to0.0pt{\hbox
to9.44447pt{\hss$\cdot$\hss}\hss}\hbox{$\mathop{\prod}$}}_{j\in\mathbbm{b}_{n}}(-|\beta_{j}|)$
and thus that $p_{n}^{*}(z)$ has the same zeros as $p_{n}^{\alpha*}(z)$, none
of which is in $\mathbb{D}$. Thus the numerator of $\phi_{n}^{*}$ will not
vanish at ${\alpha}_{n}\in\mathbb{D}$ but one of the factors
$(1-\overline{\beta}_{j}{\alpha}_{n})$ from
$\dot{\pi}_{n}^{\beta}(\alpha_{n})$ could be zero. Thus a normalization
$\phi_{n}^{*}(\alpha_{n})>0$ is not an option in general. We could however
make $s_{n}^{\alpha}=1$ when we choose
$p_{n}^{*}(\alpha_{n})/p_{n}^{\alpha*}(\alpha_{n})>0$ or, since
$\phi_{n}^{\alpha*}(\alpha_{n})>0$, this is equivalent with
$\varsigma_{n}\,p_{n}^{*}(\alpha_{n})/\pi_{n}^{\alpha}(\alpha_{n})>0$. Yet
another way to put this is requiring that
$\phi_{n}^{*}(z)/\dot{B}_{n}^{\beta}(z)$ is positive at $z=\alpha_{n}$. This
does not give a problem with 0 or $\infty$ since

$\displaystyle\dot{B}_{n}^{\alpha}(z)\phi_{n*}(z)=\frac{\phi_{n}^{*}(z)}{\dot{B}_{n}^{\beta}(z)}=\frac{\dot{\varsigma}_{n}^{\alpha}\,p_{n}^{*}(z)}{\dot{\pi}_{n}^{\alpha}(z){\hbox
to0.0pt{\hbox
to9.44447pt{\hss$\cdot$\hss}\hss}\hbox{$\mathop{\prod}$}}_{j\in\mathbbm{b}_{n}}{(z-\beta_{j})}},\qquad\dot{\varsigma}_{n}^{\alpha}=\prod_{j\in\mathbbm{a}_{n}}\sigma_{j}.$
(3.1)

It is clear that neither the numerator nor the denominator will vanish for
$z=\alpha_{n}$.

Of course a similar argument can be given if $\gamma_{n}=\beta_{n}$. Then we
choose $\phi_{n}^{*}(z)/\dot{B}^{\alpha}_{n}(z)$ to be positive at
$z=\beta_{n}$ or equivalently
$\varsigma_{n}\,p_{n}^{*}(\beta_{n})/\pi_{n}^{\beta}(\beta_{n}){\hbox
to0.0pt{\hbox
to9.44447pt{\hss$\cdot$\hss}\hss}\hbox{$\mathop{\prod}$}}_{j\in\mathbbm{a}_{n}}(-|\alpha_{j}|)>0$.

Let us formulate the result about the numerators as a lemma for further
reference.

######  Lemma 3.8.

With the normalization that we just imposed the numerators $p_{n}^{\nu}$ of
$\phi_{n}^{\nu}=p_{n}^{\nu}/\pi_{n}^{\nu}$, $\nu\in\\{\alpha,\beta,\gamma\\}$
and $n\geq 1$ are related by

$\displaystyle p_{n}(z)=p_{n}^{\alpha}(z)\mathop{{\hbox to0.0pt{\hbox
to9.44447pt{\hss$\cdot$\hss}\hss}\hbox{$\displaystyle\mathop{\prod}$}}}_{j\in\mathbbm{b}_{n}}(-|\beta_{j}|)=p_{n}^{\beta*}(z)\varsigma_{n}\mathop{{\hbox
to0.0pt{\hbox
to9.44447pt{\hss$\cdot$\hss}\hss}\hbox{$\displaystyle\mathop{\prod}$}}}_{j\in\mathbbm{a}_{n}}(-|\alpha_{j}|),\qquad\text{if}\quad\gamma_{n}=\alpha_{n}$

and

$\displaystyle p_{n}(z)=p_{n}^{\beta}(z)\mathop{{\hbox to0.0pt{\hbox
to9.44447pt{\hss$\cdot$\hss}\hss}\hbox{$\displaystyle\mathop{\prod}$}}}_{j\in\mathbbm{a}_{n}}(-|\alpha_{j}|)=p_{n}^{\alpha*}(z)\varsigma_{n}\mathop{{\hbox
to0.0pt{\hbox
to9.44447pt{\hss$\cdot$\hss}\hss}\hbox{$\displaystyle\mathop{\prod}$}}}_{j\in\mathbbm{b}_{n}}(-|\beta_{j}|),\qquad\text{if}\quad\gamma_{n}=\beta_{n},$

where as before $\varsigma_{n}=\prod_{j=1}^{n}\sigma_{j}$.

###### Proof.

The first expression for $\gamma_{n}=\alpha_{n}$ has been proved above. The
second one follows in a similar way from the relation
$\phi_{n}(z)=\phi_{n}^{\beta*}(z)\dot{B}_{n}^{\alpha}(z)$. Indeed

$\displaystyle\frac{p_{n}(z)}{\pi_{n}(z)}=\frac{\varsigma_{n}p_{n}^{\beta*}(z)}{\prod\limits_{j\in\mathbbm{a}_{n}}\varpi_{j}^{\beta}(z)\prod\limits_{j\in\mathbbm{b}_{n}}\varpi_{j}^{\beta}(z)}\prod\limits_{j\in\mathbbm{a}_{n}}\sigma_{j}\frac{z-\alpha_{j}}{1-\overline{\alpha}_{j}z}$
$\displaystyle\hphantom{\frac{p_{n}(z)}{\pi_{n}(z)}}{}=\frac{\varsigma_{n}p_{n}^{\beta*}(z)}{\prod\limits_{j\in\mathbbm{a}_{n}}\varpi_{j}^{\alpha}(z)\prod\limits_{j\in\mathbbm{b}_{n}}\varpi_{j}^{\beta}(z)}\prod\limits_{j\in\mathbbm{a}_{n}}\sigma_{j}\frac{z-\alpha_{j}}{1-\overline{\beta}_{j}z}=\frac{\varsigma_{n}p_{n}^{\beta*}(z)}{\pi_{n}(z)}\mathop{{\hbox
to0.0pt{\hbox
to9.44447pt{\hss$\cdot$\hss}\hss}\hbox{$\displaystyle\mathop{\prod}$}}}_{j\in\mathbbm{a}_{n}}\sigma_{j}(-\alpha_{j})\frac{z-\alpha_{j}}{z-\alpha_{j}}.$

With $-\sigma_{j}\alpha_{j}=-|\alpha_{j}|$ the result follows.

The case $\gamma_{n}=\beta_{n}$ is similar. ∎

Note that this normalization again means that we take the leading coefficient
of $\phi_{n}$ to be positive in the following sense. If
$\gamma_{n}=\alpha_{n}$ then
$\phi_{n}(z)=\overline{(\dot{B}_{n}^{\alpha}\phi_{n*})(\alpha_{n})}\dot{B}_{n}^{\alpha}(z)+\psi_{n-1}(z)$
with $\psi_{n-1}\in\mathcal{L}_{n-1}$, while
$\dot{B}_{n}^{\alpha}\phi_{n*}=\phi_{n}^{\alpha*}$ and
$\phi_{n}^{\alpha*}(\alpha_{n})>0$. If $\gamma_{n}=\beta_{n}$ then
$\phi_{n}(z)=\overline{(\dot{B}_{n}^{\beta}\phi_{n*})(\beta_{n})}\dot{B}_{n}^{\beta}(z)+\psi_{n-1}(z)$
with $\psi_{n-1}\in\mathcal{L}_{n-1}$ and the conclusion follows similarly.

Whenever we use the term orthonormal, we assume this normalization and
$\\{\phi_{n}\colon n=0,1,$ $2,\ldots\\}$ will denote this orthonormal system.

Thus we have proved the following Theorem. It says that if
$\gamma_{n}=\alpha_{n}$, then $\phi_{n}$ is a ‘shifted’ version of
$\phi_{n}^{\alpha}$ where ‘shifted’ means multiplied by $\dot{B}_{n}^{\beta}$:

$\displaystyle\dot{B}_{n}^{\beta}(z)\phi_{n}(z)=\dot{B}_{n}^{\beta}(z)[a_{0}B_{0}^{\alpha}+\cdots+a_{n}B_{n}^{\alpha}(z)]=a_{0}\dot{B}_{n}^{\beta}(z)+\cdots+a_{n}\dot{B}_{n}^{\alpha}(z),$

and a similar interpretation if $\gamma_{n}=\beta_{n}$. We summarize this in
the following theorem.

######  Theorem 3.9.

Assume all ORFs $\phi_{n}^{\nu}$, $\nu\in\\{\alpha,\beta,\gamma\\}$ are
orthonormal with positive leading coefficient, i.e.,

$\displaystyle\phi_{n}^{\alpha*}({\alpha}_{n})>0\qquad\mbox{and}\qquad\phi_{n}^{\beta*}({\beta}_{n})>0\qquad\text{and}\qquad\begin{cases}(\phi_{n}^{*}/\dot{B}_{n}^{\beta})(\alpha_{n})>0&\mbox{if}~{}~{}\gamma_{n}=\alpha_{n},\\\
(\phi_{n}^{*}/\dot{B}_{n}^{\alpha})(\beta_{n})>0&\mbox{if}~{}~{}\gamma_{n}=\beta_{n}.\end{cases}$

Then for all $n\geq 0$

$\displaystyle\phi_{n}=(\phi_{n}^{\alpha})\dot{B}_{n}^{\beta}=(\phi_{n}^{\beta})^{*}\dot{B}_{n}^{\alpha}\qquad\mbox{and}\qquad\phi_{n}^{*}=(\phi_{n}^{\alpha})^{*}\dot{B}_{n}^{\beta}=(\phi_{n}^{\beta})\dot{B}_{n}^{\alpha}\qquad\mbox{if~{}~{}$\gamma_{n}=\alpha_{n}$},$

while

$\displaystyle\phi_{n}=(\phi_{n}^{\beta})\dot{B}_{n}^{\alpha}=(\phi_{n}^{\alpha})^{*}\dot{B}_{n}^{\beta}\qquad\mbox{and}\qquad\phi_{n}^{*}=(\phi_{n}^{\beta})^{*}\dot{B}_{n}^{\alpha}=(\phi_{n}^{\alpha})\dot{B}_{n}^{\beta}\qquad\mbox{if~{}~{}$\gamma_{n}=\beta_{n}$}.$

######  Corollary 3.10.

We have for all $n\geq 1$ that
$(\phi_{n}^{\nu})^{*}\perp\zeta_{n}^{\nu}\mathcal{L}_{n-1}^{\nu}$,
$\nu\in\\{\alpha,\beta,\gamma\\}$.

######  Corollary 3.11.

The rational functions $\phi_{n}^{\alpha}$ and $\phi_{n}^{\alpha*}$ are in
$\mathcal{L}_{n}^{\alpha}$ and hence have all their poles in
$\\{\beta_{j}\colon j=1,\ldots,n\\}\subset\mathbb{E}$ while the zeros of
$\phi_{n}^{\alpha}$ are all in $\mathbb{D}$ and the zeros of
$\phi_{n}^{\alpha*}$ are all in $\mathbb{E}$.

The rational functions $\phi_{n}^{\beta}$ and $\phi_{n}^{\beta*}$ are in
$\mathcal{L}_{n}^{\beta}$ and hence have all their poles in
$\\{\alpha_{j}\colon j=1,\ldots,n\\}\subset\mathbb{D}$ while the zeros of
$\phi_{n}^{\beta}$ are all in $\mathbb{E}$ and the zeros of
$\phi_{n}^{\beta*}$ are all in $\mathbb{D}$.

The rational functions $\phi_{n}$ and $\phi_{n}^{*}$ are in $\mathcal{L}_{n}$
and hence have all their poles in $\\{\beta_{j}\colon
j\in\mathbbm{a}_{n}\\}\cup\\{\alpha_{j}\colon j\in\mathbbm{b}_{n}\\}$.

The zeros of $\phi_{n}$ are the same as the zeros of $\phi_{n}^{\alpha}$ and
thus are all in $\mathbb{D}$ if $\gamma_{n}=\alpha_{n}$ and they are the same
as the zeros of $\phi_{n}^{\beta}$ and thus they are all in $\mathbb{E}$ if
$\gamma_{n}=\beta_{n}$.

###### Proof.

It is well known that the zeros of $\phi_{n}^{\alpha}$ are all in $\mathbb{D}$
[9, Corollary 3.1.4], and because $\phi_{n}^{\beta}=\phi_{n*}^{\alpha}$, this
means that the zeros of $\phi_{n}^{\beta}$ are all in $\mathbb{E}$.

Because
$\phi_{n}=(\phi_{n}^{\alpha})\dot{B}_{n}^{\beta}=(\phi_{n}^{\alpha})/\prod_{j\in\mathbbm{b}_{n}}\zeta_{j}^{\alpha}$
if $\gamma_{n}=\alpha_{n}$, i.e., $n\in\mathbbm{a}_{n}$, and the product with
$\dot{B}_{n}^{\beta}$ will only exchange the poles
$1/\overline{\alpha}_{j}=\beta_{j}$, $j\in\mathbbm{b}_{n}$ in $\phi_{n}$ for
poles $\alpha_{j}=1/\overline{\beta}_{j}$, the zeros of $\phi_{n}^{\alpha}$
are left unaltered.

The proof for $n\in\mathbbm{b}_{n}$ is similar. ∎

One may summarize that for $f\in\mathcal{L}_{n}^{\nu}$ the $f_{*}$ transform
reflects both zeros and poles in $\mathbb{T}$ since $z\mapsto
z_{*}=1/\overline{z}$, while the transform $f\to f^{*}$ as it is defined in
the spaces $\mathcal{L}_{n}^{\nu}$, $\nu\in\\{\alpha,\beta,\gamma\\}$, keeps
the poles but reflects the zeros since the multiplication with the respective
factors $B_{n}^{\alpha}$, $B_{n}^{\beta}$ and
$\dot{B}_{n}^{\alpha}\dot{B}_{n}^{\beta}$ will only undo the reflection of the
poles that resulted from the $f_{*}$ operation.

##  4 Christoffel–Darboux relations and reproducing kernels

For $\nu\in\\{\alpha,\beta,\gamma\\}$, one may define the reproducing kernels
for the space $\mathcal{L}_{n}^{\nu}$. Given the ORF $\phi_{k}^{\nu}$, the
kernels are defined by

$\displaystyle
k_{n}^{\nu}(z,w)=\sum_{k=0}^{n}\phi_{k}^{\nu}(z)\overline{\phi_{k}^{\nu}(w)}.$

They reproduce $f\in\mathcal{L}_{n}^{\nu}$ by $\left\langle
k_{n}^{\nu}(\cdot,z),f\right\rangle=f(z)$.

The proof of the Christoffel–Darboux relations goes exactly like in the
classical case and we shall not repeat it here (see, e.g., [9, Theorem
3.1.3]).

######  Theorem 4.1.

The Christoffel–Darboux relations

$\displaystyle
k_{n}^{\nu}(z,w)=\frac{\phi_{n}^{\nu*}(z)\overline{\phi_{n}^{\nu*}(w)}-\zeta_{n}^{\nu}(z)\overline{\zeta_{n}^{\nu}(w)}\phi_{n}^{\nu}(z)\overline{\phi_{n}^{\nu}(w)}}{1-\zeta_{n}^{\nu}(z)\overline{\zeta_{n}^{\nu}(w)}}$
$\displaystyle\hphantom{k_{n}^{\nu}(z,w)}{}=\frac{\phi_{n+1}^{\nu*}(z)\overline{\phi_{n+1}^{\nu*}(w)}-\phi_{n+1}^{\nu}(z)\overline{\phi_{n+1}^{\nu}(w)}}{1-\zeta_{n+1}^{\nu}(z)\overline{\zeta_{n+1}^{\nu}(w)}}$

hold for $n\geq 0$, $\nu\in\\{\alpha,\beta,\gamma\\}$ and $z$, $w$ not among
the poles of $\phi_{n}^{\nu}$ and not on $\mathbb{T}$.

As an immediate consequence we have:

######  Theorem 4.2.

The following relations hold true:

$\displaystyle
k_{n}^{\alpha}(z,w)\dot{B}_{n}^{\beta}(z)\overline{\dot{B}_{n}^{\beta}(w)}=k_{n}(z,w)=k_{n}^{\beta}(z,w)\dot{B}_{n}^{\alpha}(z)\overline{\dot{B}_{n}^{\alpha}(w)}$

for $n\geq 0$ and $z,w\not\in(\mathbb{T}\cup\\{\beta_{j}\colon
j\in\mathbbm{a}_{n}\\}\cup\\{\alpha_{j}\colon j\in\mathbbm{b}_{n}\\})$.

###### Proof.

The first relation was directly shown above for the case
$\gamma_{n}=\alpha_{n}$. It also follows in the case
$\gamma_{n+1}=\alpha_{n+1}$ and using in the second CD-relation the first
expressions from Theorem 3.9 for $\phi_{n+1}$ and $\phi_{n+1}^{*}$. The
relation is thus valid independent of $\gamma_{n}=\alpha_{n}$ or
$\gamma_{n}=\beta_{n}$.

Similarly the second expression was derived before in the case
$\gamma_{n}=\beta_{n}$, but again, it also follows from the second CD-relation
and the first expressions from Theorem 3.9 for $\phi_{n+1}$ and
$\phi_{n+1}^{*}$ in the case $\gamma_{n+1}=\beta_{n+1}$. Again the relation
holds independently of $\gamma_{n}=\alpha_{n}$ or $\gamma_{n}=\beta_{n}$.

Alternatively, the second relation can also be derived from the second CD-
relation in the case $\gamma_{n+1}=\alpha_{n+1}$ but using the second
expressions from Theorem 3.9 for $\phi_{n+1}$ and $\phi_{n+1}^{*}$. ∎

Evaluation of the CD-relation in $\nu_{n}$ for $\nu\in\\{\alpha,\beta\\}$
results in another useful corollary.

######  Corollary 4.3.

For $\nu\in\\{\alpha,\beta\\}$ we have for $n\geq 0$

$\displaystyle
k_{n}^{\nu}(z,\nu_{n})=\phi_{n}^{\nu*}(z)\overline{\phi_{n}^{\nu*}(\nu_{n})}\qquad\text{and}\qquad
k_{n}^{\nu}(\nu_{n},\nu_{n})=|\phi_{n}^{\nu*}(\nu_{n})|^{2}.$

The latter corollary cannot immediately be used when $\nu=\gamma$ because
$\gamma_{n}$ could be equal to some pole of $\phi_{n}$ if it equals some
$1/\overline{\gamma}_{j}$ for $j<n$. In that case we can remove the
denominators in the CD relation and only keep the numerators. Hence setting

$\displaystyle
k_{n}(z,w)=\frac{C_{n}(z,w)}{\pi_{n}(z)\overline{\pi_{n}(w)}},\qquad\phi_{n}(z)=\frac{p_{n}(z)}{\pi_{n}(z)},\qquad\phi_{n}^{*}(z)=\frac{\varsigma_{n}p_{n}^{*}(z)}{\pi_{n}(z)},\qquad\varsigma_{n}\in\mathbb{T},$

the CD relation becomes

$\displaystyle
C_{n}(z,w)=\frac{p_{n}^{*}(z)\overline{p_{n}^{*}(w)}-\zeta_{n}(z)\overline{\zeta_{n}(w)}p_{n}(z)\overline{p_{n}(w)}}{1-\zeta_{n}(z)\overline{\zeta_{n}(w)}}$
$\displaystyle\hphantom{C_{n}(z,w)}{}=\frac{p_{n+1}^{*}(z)\overline{p_{n+1}^{*}(w)}-p_{n+1}(z)\overline{p_{n+1}(w)}}{(1-\zeta_{n+1}(z)\overline{\zeta_{n+1}(w)})\varpi_{n+1}(z)\overline{\varpi_{n+1}(w)}}.$
(4.1)

Thus, the first form gives

$\displaystyle
C_{n}(z,\gamma_{n})=p_{n}^{*}(z)\overline{p_{n}^{*}(\gamma_{n})}\qquad\text{and}\qquad
C_{n}(\gamma_{n},\gamma_{n})=|p_{n}^{*}(\gamma_{n})|^{2}.$

Evaluating a polynomial at infinity means taking its highest degree
coefficient, i.e., if $q_{n}(z)$ is a polynomial of degree $n$, then
$q_{n}(\infty)$ stands for its coefficient of $z^{n}$.

The second form of (4.1) gives for $\gamma_{n+1}\neq\infty$ and
$\gamma_{n}\neq\infty$

$\displaystyle
C_{n}(z,\gamma_{n})=\frac{p_{n+1}^{*}(z)\overline{p_{n+1}^{*}(\gamma_{n})}-p_{n+1}(z)\overline{p_{n+1}(\gamma_{n})}}{(1-\overline{\gamma}_{n}z)(1-|\gamma_{n+1}|^{2})}\qquad\text{and}$
$\displaystyle
C_{n}(\gamma_{n},\gamma_{n})=\frac{|p_{n+1}^{*}(\gamma_{n})|^{2}-|p_{n+1}(\gamma_{n})|^{2}}{(1-|\gamma_{n}|^{2})(1-|\gamma_{n+1}|^{2})}.$

Coupling the first and the second form in (4.1) gives

$\displaystyle\frac{|p_{n+1}^{*}(\gamma_{n})|^{2}-|p_{n+1}(\gamma_{n})|^{2}}{(1-|\gamma_{n}|^{2})(1-|\gamma_{n+1}|^{2})}=|p_{n}^{*}(\gamma_{n})|^{2}.$

For $\gamma_{n+1}=\infty$ and $\gamma_{n}\neq\infty$ we get

$\displaystyle
C_{n}(z,\gamma_{n})=\frac{p_{n+1}^{*}(z)\overline{p_{n+1}^{*}(\gamma_{n})}-p_{n+1}(z)\overline{p_{n+1}(\gamma_{n})}}{-(1-\overline{\gamma}_{n}z)}=p_{n}^{*}(z)\overline{p_{n}^{*}(\gamma_{n})}$

and

$\displaystyle
C_{n}(\gamma_{n},\gamma_{n})=\frac{|p_{n+1}^{*}(\gamma_{n})|^{2}-|p_{n+1}(\gamma_{n})|^{2}}{-(1-|\gamma_{n}|^{2})}=|p_{n}^{*}(\gamma_{n})|^{2}.$

If $\gamma_{n+1}=\infty$ and $\gamma_{n}=\infty$, the denominators in (4.1)
have to be replaced by 1, which gives

$\displaystyle
C_{n}(z,\gamma_{n})={p_{n+1}^{*}(z)\overline{p_{n+1}^{*}(\gamma_{n})}-p_{n+1}(z)\overline{p_{n+1}(\gamma_{n})}}=p_{n}^{*}(z)\overline{p_{n}^{*}(\gamma_{n})}$

and

$\displaystyle
C_{n}(\gamma_{n},\gamma_{n})={|p_{n+1}^{*}(\gamma_{n})|^{2}-|p_{n+1}(\gamma_{n})|^{2}}=|p_{n}^{*}(\gamma_{n})|^{2}.$

For $\gamma_{n}=\infty$ and $\gamma_{n+1}\neq\infty$ we obtain in a similar
way

$\displaystyle
C_{n}(z,\gamma_{n})=\frac{p_{n+1}^{*}(z)\overline{p_{n+1}^{*}(\gamma_{n})}-p_{n+1}(z)\overline{p_{n+1}(\gamma_{n})}}{z(1-|\gamma_{n+1}|^{2})}=p_{n}^{*}(z)\overline{p_{n}^{*}(\gamma_{n})}$

and

$\displaystyle
C_{n}(\gamma_{n},\gamma_{n})=\frac{|p_{n+1}^{*}(\gamma_{n})|^{2}-|p_{n+1}(\gamma_{n})|^{2}}{(1-|\gamma_{n+1}|^{2})}=|p_{n}^{*}(\gamma_{n})|^{2}.$

To summarize, the relations of Corollary 4.3 may not hold for the ORF if
$\nu=\gamma$, but similar relations do hold for the numerators as stated in
the next corollary.

######  Corollary 4.4.

If $C_{n}(z,w)$ is the numerator in the CD relation and $p_{n}(z)$ is the
numerator of the ORF $\phi_{n}$ for the sequence
$\underline{\mbox{\boldmath{$\gamma$}}}$ then we have for $n\geq 0$

$\displaystyle
C_{n}(z,\gamma_{n})=p_{n}^{*}(z)\overline{p_{n}^{*}(\gamma_{n})}\qquad\mbox{and}\qquad
C_{n}(\gamma_{n},\gamma_{n})=|p_{n}^{*}(\gamma_{n})|^{2}.$

##  5 Recurrence relation

The recurrence for the $\phi_{n}^{\alpha}$ is well known. For a proof see,
e.g., [9, Theorem 4.1.1]. For $\phi_{n}^{\beta}$ the proof can be copied by
symmetry. However, also for $\nu=\gamma$ the same recurrence and its proof can
be copied, with the exception that the derivation fails when
$p_{n}^{*}(\gamma_{n-1})=0$ where $p_{n}=\phi_{n}\pi_{n}$. This can (only)
happen if $(1-|\gamma_{n}|)(1-|\gamma_{n-1}|)<0$ (i.e., one of these
$\gamma$’s is in $\mathbb{D}$ and the other is in $\mathbb{E}$). We shall say
that $\phi_{n}$ is regular if $p_{n}^{*}(\gamma_{n-1})\neq 0$. If $\nu=\alpha$
or $\nu=\beta$ then the whole sequence $(\phi_{n}^{\nu})_{n\geq 0}$ will be
automatically regular. Thus we have the following theorem:

######  Theorem 5.1.

Let $\nu\in\\{\alpha,\beta,\gamma\\}$ and if $\nu=\gamma$ assume moreover that
$\phi_{n}^{\nu}$ is regular, then the following recursion holds with initial
condition $\phi_{0}^{\nu}=\phi_{0}^{\nu*}=1$

$\displaystyle\left[\begin{matrix}\phi_{n}^{\nu}(z)\\\
\phi_{n}^{\nu*}(z)\end{matrix}\right]=H_{n}^{\nu}\frac{\varpi_{n-1}^{\nu}(z)}{\varpi_{n}^{\nu}(z)}\left[\begin{matrix}1&{\lambda}_{n}^{\nu}\\\
\overline{\lambda}_{n}^{\nu}&1\end{matrix}\right]\left[\begin{matrix}\zeta_{n-1}^{\nu}(z)&0\\\
0&1\end{matrix}\right]\left[\begin{matrix}\phi_{n-1}^{\nu}(z)\\\
\phi_{n-1}^{\nu*}(z)\end{matrix}\right],$

where $H_{n}^{\nu}$ is a nonzero constant times a unitary matrix:

$\displaystyle H_{n}^{\nu}=e_{n}^{\nu}\left[\begin{matrix}\eta_{n1}^{\nu}&0\\\
0&\eta_{n2}^{\nu}\end{matrix}\right],\qquad
e_{n}^{\nu}\in\mathbb{C}\setminus\\{0\\},\qquad\eta_{n1}^{\nu},\eta_{n2}^{\nu}\in\mathbb{T}.$

The constant $\eta_{n1}^{\nu}$ is chosen such that the normalization condition
for the ORFs is maintained. The other constant $\eta_{n2}^{\nu}$ is then
automatically related to $\eta_{n1}^{\nu}$ by
$\eta_{n2}^{\nu}=\overline{\eta}_{n1}^{\nu}\overline{\sigma}_{n-1}\sigma_{n}$.
The Szegő parameter $\lambda_{n}^{\nu}$ is given by

$\displaystyle\lambda_{n}^{\nu}=\eta_{n}^{\nu}\frac{p_{n}^{\nu}(\nu_{n-1})}{\overline{p_{n}^{\nu*}(\nu_{n-1})}}\qquad\mbox{with}\qquad\eta_{n}^{\nu}=\overline{\varsigma}_{n-2}\frac{\overline{p_{n-1}^{\nu*}(\nu_{n-1})}}{p_{n-1}^{\nu*}(\nu_{n-1})}\in\mathbb{T},$

where $\phi_{n}^{\nu}(z)=p_{n}^{\nu}(z)/\pi_{n}^{\nu}(z)$.

###### Proof.

We immediately concentrate on the general situation $\nu=\gamma$. Of course
$\nu=\alpha$ and $\nu=\beta$ will drop out as special cases. For simplicity
assume that $\gamma_{n}$ and $\gamma_{n-1}$ are not 0 and not $\infty$. The
technicalities when this is not true are left as an exercise. It is easy
because formally it follows the steps of the proof below but one has to
replace a linear factor involving infinity by the coefficient of infinity
(like $1-\infty z=-z$ and $z-\infty=-1$) and evaluating a polynomial at
$\infty$ means taking its leading coefficient.

First we show that there are some numbers $c_{n}$ and $d_{n}$ such that

$\displaystyle\phi(z):=\frac{1-\overline{\gamma}_{n}z}{z-\gamma_{n-1}}\phi_{n}(z)-d_{n}\phi_{n-1}(z)-c_{n}\frac{1-\overline{\gamma}_{n-1}z}{z-\gamma_{n-1}}\phi_{n-1}^{*}\in\mathcal{L}_{n-2}.$

This can be written as $N(z)/[(z-\gamma_{n-1})\pi_{n-1}(z)]$. Thus the $c_{n}$
and $d_{n}$ are defined by the conditions
$N(\gamma_{n-1})=N(1/\overline{\gamma}_{n-1})=0$. If we denote
$\phi_{k}=p_{k}/\pi_{k}$ and thus
$\phi_{k}^{*}=p_{k}^{*}\varsigma_{k}/\pi_{k}$, it is clear that

$\displaystyle
N(z)=p_{n}(z)-d_{n}(z-\gamma_{n-1})p_{n-1}(z)-c_{n}(1-\overline{\gamma}_{n-1}z)p_{n-1}^{*}(z)\varsigma_{n-1}.$

Thus the first condition gives

$\displaystyle
c_{n}=\frac{\overline{\varsigma}_{n-1}p_{n}(\gamma_{n-1})}{(1-|\gamma_{n-1}|^{2})p_{n-1}^{*}(\gamma_{n-1})}$

and the second one

$\displaystyle
d_{n}=\frac{p_{n}(1/\overline{\gamma}_{n-1})}{(1/\overline{\gamma}_{n-1}-\gamma_{n-1})p_{n-1}(1/\overline{\gamma}_{n-1})}=\frac{\overline{p_{n}^{*}(\gamma_{n-1})}}{(1-|\gamma_{n-1}|^{2})\overline{p_{n-1}^{*}(\gamma_{n-1})}}.$

Note that $p_{n-1}^{*}(\gamma_{n-1})$ cannot be zero by Corollary 3.11, and
that also $p_{n}^{*}(\gamma_{n-1})$ does not vanish by our assumption of
regularity.

Furthermore, by using the orthognality of $\phi_{n}$ and $\phi_{n-1}$ and
Corollary 3.10, it is not difficult to show that $\phi\perp\mathcal{L}_{n-2}$
so that it must be identically zero. Thus

$\displaystyle\phi_{n}(z)=d_{n}\overline{\sigma}_{n-1}\frac{1-\overline{\gamma}_{n-1}z}{1-\overline{\gamma}_{n}z}[\zeta_{n-1}(z)\phi_{n-1}(z)+\lambda_{n}\phi_{n-1}^{*}(z)],$

with

$\displaystyle\lambda_{n}=\eta_{n}\frac{p_{n}(\gamma_{n-1})}{\overline{p_{n}^{*}(\gamma_{n-1})}},\qquad\eta_{n}=\overline{\varsigma}_{n-1}\sigma_{n-1}\frac{\overline{p_{n-1}^{*}(\gamma_{n-1})}}{p_{n-1}^{*}(\gamma_{n-1})}.$

By taking the $(\ )^{*}$ transform (in $\mathcal{L}_{n}$) we obtain

$\displaystyle\phi_{n}^{*}(z)=\overline{d}_{n}{\sigma}_{n}\frac{1-\overline{\gamma}_{n-1}z}{1-\overline{\gamma}_{n}z}[\overline{\lambda}_{n}\zeta_{n-1}(z)\phi_{n-1}(z)+\phi_{n-1}^{*}(z)].$

This proves the recurrence by taking $e_{n}=|d_{n}|$ and
$\overline{\eta}_{n1}={\sigma}_{n-1}\mathfrak{u}({d}_{n})$.

It remains to show that the initial step for $n=1$ is true. Since
$\phi_{0}=\phi_{0}^{*}=1$, then in case $\gamma_{0}=\alpha_{0}=0$, hence
$\zeta_{0}=z$, we have

$\displaystyle\phi_{1}(z)=e_{1}\eta_{11}\frac{z+\lambda_{1}}{1-\overline{\gamma}_{1}z}\qquad\text{and}\qquad\phi_{1}^{*}(z)=e_{1}{\eta}_{12}\frac{\overline{\lambda}_{1}z+1}{1-\overline{\gamma}_{1}z}.$

Thus

$\displaystyle p_{1}(z)=e_{1}\eta_{11}(z+\lambda_{1})\qquad\text{and}\qquad
p_{1}^{*}(z)=e_{1}\overline{\eta}_{11}(\overline{\lambda}_{1}z+1).$

This implies that $\lambda_{1}$ is indeed given by the general formula because

$\displaystyle\lambda_{1}=\eta_{1}\frac{p_{1}(\gamma_{0})}{\overline{p_{1}^{*}(\gamma_{0})}}=\frac{p_{1}(0)}{\overline{p_{1}^{*}(0)}}=\frac{e_{1}\eta_{11}\lambda_{1}}{e_{1}\eta_{11}}.$

In case $\gamma_{0}=\beta_{0}=\infty$, then $\zeta_{0}=1/z$, so that

$\displaystyle\phi_{1}(z)=e_{1}\eta_{11}\frac{-1-\lambda_{1}z}{1-\overline{\gamma}_{1}z}\qquad\text{and}\qquad\phi_{1}^{*}(z)=e_{1}{\eta}_{12}\frac{-\overline{\lambda}_{1}-z}{1-\overline{\gamma}_{1}z},$

and thus

$\displaystyle p_{1}(z)=-e_{1}\eta_{11}(1+\lambda_{1}z)\qquad\text{and}\qquad
p_{1}^{*}(z)=-e_{1}\overline{\eta}_{11}\overline{\sigma}_{1}(\overline{\lambda}_{1}+z),$

and again $\lambda_{1}$ is given by the general formula

$\displaystyle\lambda_{1}=\eta_{1}\frac{p_{1}(\gamma_{0})}{\overline{p_{1}^{*}(\gamma_{0})}}=1\frac{p_{1}(\infty)}{\overline{p_{1}^{*}(\infty)}}=\frac{e_{1}\eta_{11}\lambda_{1}}{e_{1}\eta_{11}}.$

This proves the theorem. ∎

######  Remark 5.2.

If $\nu\in\\{\alpha,\beta\\}$ we could rewrite $\lambda_{n}^{\nu}$ in terms of
$\phi_{n}^{\nu}$ because by dividing and multiplying with the appropriate
denominators $\pi_{n}^{\nu}$ one gets

$\displaystyle\lambda_{n}^{\nu}=\eta_{n}^{\nu}\frac{\phi_{n}^{\nu}(\nu_{n-1})}{\overline{\phi_{n}^{\nu*}(\nu_{n-1})}},\qquad\eta_{n}^{\nu}=\sigma_{n-1}\overline{\sigma}_{n}\frac{(1-\overline{\nu}_{n}\nu_{n-1})}{(1-\nu_{n}\overline{\nu}_{n-1})}\frac{\overline{\phi_{n-1}^{\nu*}(\nu_{n-1})}}{\phi_{n-1}^{\nu*}(\nu_{n-1})},\qquad
n\geq 1.$

Note that also this $\eta_{n}^{\nu}\in\mathbb{T}$, but it differs from the
$\eta_{n}^{\nu}$ in the previous theorem. However if $\nu=\gamma$, then this
expression has the disadvantage that $\gamma_{n-1}$ could be equal to
$1/\overline{\gamma}_{n}$ or it could be equal to a pole of $\phi_{n}$ in
which case it would not make sense to evaluate these expressions in
$\gamma_{n-1}$. The latter expressions only make sense if we interpret them as
limiting values

$\displaystyle\frac{{\phi_{n}^{\nu}(\nu_{n-1})}}{\overline{\phi_{n}^{\nu*}(\nu_{n-1})}}=\lim_{z\to\nu_{n-1}}\frac{{\phi_{n}^{\nu}(z)}}{\overline{\phi_{n}^{\nu*}(z)}}\qquad\text{and}$
$\displaystyle\frac{(1-\overline{\nu}_{n}\nu_{n-1})}{(1-\nu_{n}\overline{\nu}_{n-1})}\frac{\overline{\phi_{n-1}^{\nu*}(\nu_{n-1})}}{\phi_{n-1}^{\nu*}(\nu_{n-1})}=\lim_{z\to\nu_{n-1}}\frac{(1-\overline{\nu}_{n}z)}{(1-\nu_{n}\overline{z})}\frac{\overline{\phi_{n-1}^{\nu*}(z)}}{\phi_{n-1}^{\nu*}(z)},$

where one has to assume that $\lim\limits_{\xi\to 0}[\xi/\overline{\xi}]=1$.
We shall from now on occasionally use these expressions with this
interpretation, but the expressions for $\lambda_{n}^{\nu}$ from Theorem 5.1
using the numerators are more direct since they immediately give the limiting
values.

Note that $\lambda_{n}^{\alpha}$ is the value of a Blaschke product with all
its zeros in $\mathbb{D}$ evaluated at $\alpha_{n-1}\in\mathbb{D}$ and
therefore $\lambda_{n}^{\alpha}\in\mathbb{D}$. Similarly,
$\lambda_{n}^{\beta}$ is the value of a Blaschke product with all its zeros in
$\mathbb{E}$, evaluated at $\beta_{n-1}\in\mathbb{E}$ so that
$\lambda_{n}^{\beta}\in\mathbb{D}$. Since the zeros of $\phi_{n}$ are the
zeros of $\phi_{n}^{\alpha}$ if $n\in\mathbbm{a}_{n}$ and they are the zeros
of $\phi_{n}^{\beta}$ if $n\in\mathbbm{b}_{n}$, it follows that if $n$ and
$n-1$ are both in $\mathbbm{a}_{n}$ or both in $\mathbbm{b}_{n}$, then
$\lambda_{n}\in\mathbb{D}$ but if $n\in\mathbbm{a}_{n}$ and
$n-1\in\mathbbm{b}_{n}$ or vice versa, then $\lambda_{n}\in\mathbb{E}$.
Therefore

$\displaystyle(e_{n}^{\nu})^{2}=\frac{1-|\nu_{n}|^{2}}{1-|\nu_{n-1}|^{2}}\frac{1}{1-|\lambda_{n}^{\nu}|^{2}}>0$
(5.1)

and we can choose $e_{n}$ as the positive square root of this expression. The
above expression is derived in [9, Theorem 4.1.2] for the case $\nu=\alpha$ by
using the CD relations. By symmetry, this also holds for $\nu=\beta$. For
$\nu=\gamma$, the same relation can be obtained by stripping the denominators
as we explained after the proof of the CD-relation in Section 4.

What goes wrong with the recurrence relation when $\phi_{n}$ is not regular?
From the proof of Theorem 5.1, it follows that then $d_{n}=0$. We still have
the relation

$\displaystyle\phi_{n}(z)=\frac{\overline{\sigma}_{n-1}}{(1-|\gamma_{n-1}|^{2})\overline{p_{n-1}^{*}(\gamma_{n-1})}}\left[p_{n}^{*}(\gamma_{n-1})\zeta_{n-1}(z)\phi_{n-1}(z)+s_{n-1}p_{n}(\gamma_{n-1})\phi_{n-1}^{*}(z)\right]$

with
$s_{n-1}=\frac{\varsigma_{n-1}p_{n-1}^{*}(\gamma_{n-1})}{\overline{\sigma}_{n-1}\overline{p_{n-1}^{*}(\gamma_{n-1})}}\in\mathbb{T}$
and $p_{n}^{*}(\gamma_{n-1})=0$. Thus, there is some positive constant $e_{n}$
and some $\eta_{n1}\in\mathbb{T}$ such that

$\displaystyle\phi_{n}(z)=e_{n}\eta_{n1}\frac{\varpi_{n-1}(z)}{\varpi_{n}(z)}\big{[}0\;\zeta_{n-1}(z)\phi_{n-1}(z)+\phi_{n-1}^{*}(z)\big{]},$

i.e., the first term in the sum between square brackets vanishes. Applying
Theorem 5.1 in this case would give $\lambda_{n}=\infty$, and the previous
relations show that we only have to replace in Theorem 5.1 the matrix

$\displaystyle\left[\begin{matrix}1&\lambda_{n}^{\nu}\\\
\overline{\lambda_{n}^{\nu}}&1\end{matrix}\right]\qquad\text{by}\qquad\left[\begin{matrix}0&1\\\
1&0\end{matrix}\right].$

This is in line with how we have dealt with $\infty$ so far where the rule of
thumb was to set $a-\nu b=-b$ if $\nu=\infty$. So let us therefore also use
Theorem 5.1 with this interpretation when $\phi_{n}$ is not regular and thus
$\lambda_{n}=\infty$. With the expressions at the end of Section 4, it can
also be shown that in this case

$\displaystyle e_{n}^{2}=-\frac{1-|\gamma_{n}|^{2}}{1-|\gamma_{n-1}|^{2}}>0.$

Note that this corresponds to replacing $1-|\lambda_{n}|^{2}$ when
$\lambda_{n}=\infty$ by $-1$. Since this non-regular situation can only occur
when $(1-|\gamma_{n}|)(1-|\gamma_{n-1}|)<0$, this expression for $e_{n}^{2}$
is indeed positive. A similar rule can be applied if $\gamma_{n}$ or
$\gamma_{n-1}$ is infinite, just replace in this or previous expression
$1-|\infty|^{2}$ by $-1$. The positivity of the expressions for $e_{n}^{2}$
also follows from the following result.

######  Theorem 5.3.

The Szegő parameters satisfy for all $n\geq 1$:

If $\gamma_{n}=\alpha_{n}$ and $\gamma_{n-1}=\alpha_{n-1}$ then
$\lambda_{n}=\lambda_{n}^{\alpha}=\overline{\lambda_{n}^{\beta}}\in\mathbb{D}$.

If $\gamma_{n}=\beta_{n}$ and $\gamma_{n-1}=\beta_{n-1}$ then
$\lambda_{n}=\lambda_{n}^{\beta}=\overline{\lambda_{n}^{\alpha}}\in\mathbb{D}$.

If $\gamma_{n}=\alpha_{n}$ and $\gamma_{n-1}=\beta_{n-1}$ then
$\lambda_{n}=1/\overline{\lambda_{n}^{\beta}}=1/\lambda_{n}^{\alpha}\in\mathbb{E}$.

If $\gamma_{n}=\beta_{n}$ and $\gamma_{n-1}=\alpha_{n-1}$ then
$\lambda_{n}=1/\overline{\lambda_{n}^{\alpha}}=1/\lambda_{n}^{\beta}\in\mathbb{E}$.

###### Proof.

Suppose $\gamma_{n}=\alpha_{n}$ and $\gamma_{n-1}=\alpha_{n-1}$, then by
Theorems 5.1 and 3.9, or better still by Lemma 3.8,

$\displaystyle\lambda_{n}=\left(\overline{\varsigma}_{n-2}\frac{\overline{p_{n-1}^{*}(\alpha_{n-1})}}{p_{n-1}^{*}(\alpha_{n-1})}\right)\;\frac{p_{n}(\alpha_{n-1})}{\overline{p_{n}^{*}(\alpha_{n-1})}}=\left(\overline{\varsigma}_{n-2}\frac{\overline{p_{n-1}^{\alpha*}(\alpha_{n-1})}}{p_{n-1}^{\alpha*}(\alpha_{n-1})}\right)\;\frac{p_{n}^{\alpha}(\alpha_{n-1})}{\overline{p_{n}^{\alpha*}(\alpha_{n-1})}}=\lambda_{n}^{\alpha}.$

When using $p_{n}^{\alpha}(z)=\varsigma_{n}\,p_{n}^{\beta*}(z)\mathop{{\hbox
to0.0pt{\hbox
to9.44447pt{\hss$\cdot$\hss}\hss}\hbox{$\mathop{\prod}$}}}_{j=1}^{n}(-|\alpha_{j}|)$
and $\alpha_{j}=1/\overline{\beta}_{j}$, the previous relation becomes

$\displaystyle\lambda_{n}=\left(\overline{\varsigma}_{n-2}\frac{\varsigma_{n-1}\overline{p_{n-1}^{\beta}(1/\overline{\beta}_{n-1})}}{\overline{\varsigma}_{n-1}p_{n-1}^{\beta}(1/\overline{\beta}_{n-1})}\right)\frac{p_{n}^{\beta*}(1/\overline{\beta}_{n-1})}{\overline{p_{n}^{\beta}(1/\overline{\beta}_{n-1})}}$
$\displaystyle\hphantom{\lambda_{n}}{}=\left(\sigma_{n-1}^{2}\varsigma_{n-2}\frac{p_{n}^{\beta*}(\beta_{n-1})}{\overline{p_{n}^{\beta*}(\beta_{n-1})}}\frac{\overline{\beta}_{n-1}^{n-1}}{\beta_{n-1}^{n-1}}\right)\frac{\overline{p_{n}^{\beta}(\beta_{n-1})}}{p_{n}^{\beta*}(\beta_{n-1})}\frac{\beta_{n-1}^{n}}{\overline{\beta}_{n-1}^{n}}$
$\displaystyle\hphantom{\lambda_{n}}{}=\sigma_{n-1}^{2}\frac{\beta_{n-1}}{\overline{\beta}_{n-1}}\overline{\lambda_{n}^{\beta}}=\overline{\lambda_{n}^{\beta}}.$

The proof for $\gamma_{n}=\beta_{n}$ and $\gamma_{n-1}=\beta_{n-1}$, $n\geq 1$
is similar.

Next consider $\gamma_{n}=\alpha_{n}$ and $\gamma_{n-1}=\beta_{n-1}$, then

$\displaystyle\lambda_{n}=\left(\overline{\varsigma}_{n-2}\frac{\overline{p_{n-1}^{\beta*}(\beta_{n-1})}}{p_{n-1}^{\beta*}(\beta_{n-1})}\right)\frac{p_{n}^{\alpha}(\beta_{n-1})}{\overline{p_{n}^{\alpha*}(\beta_{n-1})}}=\eta_{n}^{\beta}\frac{p_{n}^{\beta*}(\beta_{n-1})}{\overline{p_{n}^{\beta}(\beta_{n-1})}}=\frac{\eta_{n}^{\beta}\overline{\eta_{n}^{\beta}}}{\overline{\lambda_{n}^{\beta}}}=\frac{1}{\overline{\lambda_{n}^{\beta}}}=\frac{1}{\lambda_{n}^{\alpha}}.$

The remaining case $\gamma_{n}=\beta_{n}$ and $\gamma_{n-1}=\alpha_{n-1}$, is
again similar. ∎

######  Remark 5.4.

It should also be clear that the expression for the parameters
$\lambda_{n}^{\nu}$ of Theorem 5.1 are for theoretical use only. They are
expressed in terms of $p_{n}^{\nu}$, which is the numerator of
$\phi_{n}^{\nu}$, the object that should be the result of the computation, and
hence unknown at the moment of computing $\lambda_{n}^{\nu}$. For practical
use, these parameters $\lambda_{n}^{\nu}$ should be obtained in a different
way. In inverse eigenvalue problems the recursion is used backwards, i.e., for
decreasing degrees of the ORFs and then these expressions can be used of
course.

Even if we know the $\lambda_{n}^{\nu}$, then in Theorem 5.1, we still need
the normalizing factor $\eta_{n1}^{\nu}\in\mathbb{T}$ which is characterized
by “$\eta_{n1}^{\nu}$ is chosen such that the normalization condition or the
ORFs is maintained”. We could compute $\phi_{n}^{\nu}$ with
$\eta_{n1}^{\nu}=1$, check the value of $\phi_{n}^{\nu*}(\gamma_{n-1})$, and
derive $\eta_{n1}^{\nu}$ from it. What this means is shown in the next lemma.

######  Lemma 5.5.

For $\nu\in\\{\alpha,\beta\\}$, the phase $\theta_{n}^{\nu}$ of the unitary
factor $\eta_{n1}^{\nu}=e^{i\theta_{n}^{\nu}}$ is given by

$\displaystyle\theta_{n}^{\nu}=\mathrm{arg}\bigl{(}\overline{\sigma}_{n-1}\sigma_{n}\overline{\varpi^{\nu}_{n}(\nu_{n-1})}\overline{\phi_{n}^{\nu*}(\nu_{n-1})}\bigr{)}$

or equivalently

$\displaystyle\eta_{n1}^{\nu}=\overline{\sigma}_{n-1}\sigma_{n}\mathfrak{u}\bigl{(}\varpi_{n}^{\nu}(\nu_{n-1})\phi_{n}^{\nu*}(\nu_{n-1})\bigr{)}.$

$($Recall $\mathfrak{u}(z)=\overline{z}/|z|$.$)$

###### Proof.

Take the first relation of Theorem 5.1 and evaluate for $z=\nu_{n-1}$, then
because$\varpi_{n-1}^{\nu*}(\nu_{n-1})=0$ we get

$\displaystyle\phi_{n}^{\nu}(\nu_{n-1})\varpi_{n}^{\nu}(\nu_{n-1})=e_{n}^{\nu}\eta_{n1}^{\nu}\varpi_{n-1}^{\nu}(\nu_{n-1})\lambda_{n}^{\nu}\phi_{n-1}^{\nu*}(\nu_{n-1})$

or

$\displaystyle\eta_{n1}^{\nu}=\frac{\varpi_{n}^{\nu}(\nu_{n-1})\phi_{n}^{\nu}(\nu_{n-1})}{e_{n}^{\nu}\varpi_{n-1}^{\nu}(\nu_{n-1})\lambda_{n}^{\nu}\phi_{n-1}^{\nu*}(\nu_{n-1})}.$

Use the definition of
$\lambda_{n}^{\nu}=\eta_{n}^{\nu}\frac{\phi_{n}^{\nu}(\nu_{n-1})}{\overline{\phi_{n}^{\nu*}(\nu_{n-1})}}$
with
$\eta_{n}^{\nu}=\sigma_{n-1}\overline{\sigma}_{n}\frac{\varpi_{n}^{\nu}(\nu_{n-1})}{\varpi_{n-1}^{\nu}(\nu_{n})}$
and knowing that $\phi_{n-1}^{\nu*}(\nu_{n-1})>0$ we obtain after
simplification and leaving out all the factors with phase zero

$\displaystyle\theta_{n}^{\nu}=\mathrm{arg}\big{(}\overline{\sigma}_{n-1}\sigma_{n}\overline{\varpi^{\nu}_{n}(\nu_{n-1})}\overline{\phi_{n}^{\nu*}(\nu_{n-1})}\big{)}$

as claimed. ∎

Note that this expression for $\eta_{n1}^{\nu}$ is well defined because
$\phi_{n}^{\nu*}(\nu_{n-1})\neq 0$ if $\nu\in\\{\alpha,\beta\\}$.

For $\nu=\gamma$, the expression is a bit more involved but it can be obtained
in a similar way from the normalization conditions given in Theorem 3.9. We
skip the details.

Another solution of the recurrence relation is formed by the functions of the
second kind. Like in the classical case (i.e., for $\nu=\alpha$) we can
introduce them for $\nu\in\\{\alpha,\beta,\gamma\\}$ by (see [9, p. 83])

$\displaystyle\psi_{n}^{\nu}(z)=\int_{\mathbb{T}}[E(t,z)\phi_{n}^{\nu}(t)-D(t,z)\phi_{n}^{\nu}(z)]\mathrm{d}\mu(t).$

where $D(t,z)=\frac{t+z}{t-z}$ and $E(t,z)=D(t,z)+1=\frac{2t}{t-z}$. This
results in

$\displaystyle\psi_{0}^{\nu}=1\qquad\mbox{and}\qquad\psi_{n}^{\nu}(z)=\int_{\mathbb{T}}D(t,z)[\phi_{n}^{\nu}(t)-\phi_{n}^{\nu}(z)]\mathrm{d}\mu(t),\qquad
n\geq 1,$

which may be generalized to (see [9, Lemma 4.2.2])

$\displaystyle\psi_{n}^{\nu}(z)f(z)=\int_{\mathbb{T}}D(t,z)[\phi_{n}^{\nu}(t)f(t)-\phi_{n}^{\nu}(z)f(z)]\mathrm{d}\mu(t),\qquad
n\geq 1$

with $f$ arbitrary in $\mathcal{L}_{(n-1)*}^{\nu}$. It also holds that (see
[9, Lemma 4.2.3])

$\displaystyle\psi_{n}^{\nu*}(z)g(z)=\int_{\mathbb{T}}D(t,z)[\phi_{n}^{\nu*}(t)g(t)-\phi_{n}^{\nu*}(z)g(z)]\mathrm{d}\mu(t),\qquad
n\geq 1$

with $g$ arbitrary in $\mathcal{L}_{n*}^{\nu}(\nu_{n*})$. Recall that
$\mathcal{L}_{n*}^{\nu}(\nu_{n*})$ is the space of all functions in
$\mathcal{L}_{n*}^{\nu}$ that vanish for $z=\nu_{n*}=1/\overline{\nu}_{n}$.
This space is spanned by $\\{B_{k}^{\nu}/B_{n}^{\nu}\colon k=0,\ldots,n-1\\}$
if $\nu\in\\{\alpha,\beta\\}$. For $\nu=\gamma$, the space can be
characterized as (see Lemma 3.1)

$\displaystyle\mathcal{L}_{n*}(\gamma_{n*})=\mathop{\rm
span}\nolimits\left\\{\frac{B_{k}}{\dot{B}_{n}^{\alpha}\dot{B}_{n}^{\beta}}\right\\}_{k=0}^{n-1}=\mathop{\rm
span}\nolimits\left\\{\frac{B_{k}^{\alpha}}{\zeta_{n}\dot{B}_{n-1}^{\alpha}}\right\\}_{k=0}^{n-1}=\mathop{\rm
span}\nolimits\left\\{\frac{B_{k}^{\beta}}{\zeta_{n}\dot{B}_{n-1}^{\beta}}\right\\}_{k=0}^{n-1}.$

######  Theorem 5.6.

The following relations for the functions of the second kind hold for $n\geq
0$:

$\displaystyle\psi_{n}=(\psi_{n}^{\alpha})\dot{B}_{n}^{\beta}=(\psi_{n}^{\beta})^{*}\dot{B}_{n}^{\alpha}\qquad\mbox{and}\qquad\psi_{n}^{*}=(\psi_{n}^{\alpha})^{*}\dot{B}_{n}^{\beta}=(\psi_{n}^{\beta})\dot{B}_{n}^{\alpha}\qquad\mbox{if~{}~{}$\gamma_{n}=\alpha_{n}$},$

while

$\displaystyle\psi_{n}=(\psi_{n}^{\beta})\dot{B}_{n}^{\alpha}=(\psi_{n}^{\alpha})^{*}\dot{B}_{n}^{\beta}\qquad\mbox{and}\qquad\psi_{n}^{*}=(\psi_{n}^{\beta})^{*}\dot{B}_{n}^{\alpha}=(\psi_{n}^{\alpha})\dot{B}_{n}^{\beta}\qquad\mbox{if~{}~{}$\gamma_{n}=\beta_{n}$}.$

We assume the normalization of Theorem 3.9.

###### Proof.

This is trivial for $n=0$, hence suppose $n\geq 1$ and $\gamma_{n}=\alpha_{n}$
then

$\displaystyle\psi_{n}(z)=\int_{\mathbb{T}}D(t,z)[\phi_{n}(t)-\phi_{n}(z)]\mathrm{d}\mu(t)=\int_{\mathbb{T}}D(t,z)[\phi_{n}^{\alpha}(t)\dot{B}_{n}^{\beta}(t)-\phi_{n}^{\alpha}(z)\dot{B}_{n}^{\beta}(z)]\mathrm{d}\mu(t)$
$\displaystyle\hphantom{\psi_{n}(z)}{}=\psi_{n}^{\alpha}(z)\dot{B}_{n}^{\beta}(z),$

because

$\displaystyle\dot{B}_{n}^{\beta}(z)=\dot{B}_{n-1}^{\beta}(z)=\prod_{j\in\mathbbm{b}_{n-1}}\zeta_{j}^{\beta}(z)=\prod_{j\in\mathbbm{b}_{n-1}}\zeta_{j*}^{\alpha}(z)\in\mathcal{L}_{(n-1)*}^{\alpha}.$

Moreover, using $\phi_{n*}^{\alpha}=\phi_{n}^{\beta}$ we also have

$\displaystyle\psi_{n}(z)=\int_{\mathbb{T}}D(t,z)[\phi_{n}(t)-\phi_{n}(z)]\mathrm{d}\mu(t)=\int_{\mathbb{T}}D(t,z)[\phi_{n}^{\alpha}(t)\dot{B}_{n}^{\beta}(t)-\phi_{n}^{\alpha}(z)\dot{B}_{n}^{\beta}(z)]\mathrm{d}\mu(t)$
$\displaystyle\hphantom{\psi_{n}(z)}{}=\int_{\mathbb{T}}D(t,z)[\phi_{n*}^{\beta}(t)\dot{B}_{n}^{\beta}(t)-\phi_{n*}^{\beta}(z)\dot{B}_{n}^{\beta}(z)]\mathrm{d}\mu(t)$
$\displaystyle\hphantom{\psi_{n}(z)}{}=\int_{\mathbb{T}}D(t,z)[\phi_{n}^{\beta*}(t)\dot{B}_{n}^{\alpha}(t)-\phi_{n}^{\beta*}(z)\dot{B}_{n}^{\alpha}(z)]\mathrm{d}\mu(t),$

and since $\dot{B}_{n}^{\alpha}\in\mathcal{L}_{n*}^{\beta}(\beta_{n*})$, we
also get the second part:
$\psi_{n}=(\psi_{n}^{\beta})^{*}\dot{B}_{n}^{\alpha}$.

Moreover
$\psi_{n}^{*}=[\psi_{n}^{\alpha}\dot{B}_{n}^{\beta}]_{*}\dot{B}_{n}^{\alpha}\dot{B}_{n}^{\beta}=\psi_{n*}^{\alpha}\dot{B}_{n}^{\alpha}\dot{B}_{n}^{\beta}\dot{B}_{n*}^{\beta}=[\psi_{n*}^{\alpha}\dot{B}_{n}^{\alpha}/\dot{B}_{n}^{\beta}]\dot{B}_{n}^{\beta}=\psi_{n}^{\alpha*}\dot{B}_{n}^{\beta}$.
It follows in a similar way that
$\psi_{n}^{*}=\psi_{n}^{\beta}\dot{B}_{n}^{\alpha}$.

The case $\gamma_{n}=\beta_{n}$ is proved similarly. ∎

With these relations, it is not difficult to mimic the arguments of [9,
Theorem 4.2.4] and obtain the following.

######  Theorem 5.7.

These functions satisfy the recurrence relation

$\displaystyle\left[\begin{matrix}\psi_{n}^{\nu}(z)\\\
-\psi_{n}^{\nu*}(z)\end{matrix}\right]=H_{n}^{\nu}\frac{\varpi_{n-1}^{\nu}(z)}{\varpi_{n}^{\nu}(z)}\left[\begin{matrix}1&{\lambda}_{n}^{\nu}\\\
\overline{\lambda}_{n}^{\nu}&1\end{matrix}\right]\left[\begin{matrix}\zeta_{n-1}^{\nu}(z)&0\\\
0&1\end{matrix}\right]\left[\begin{matrix}\psi_{n-1}^{\nu}(z)\\\
-\psi_{n-1}^{\nu*}(z)\end{matrix}\right],\qquad n\geq 1$

with $\psi_{0}^{\nu}=\psi_{0}^{\nu*}=1$ and all other quantities as in Theorem
5.1.

##  6 Para-orthogonal rational functions

Before we move on to quadrature formulas, we define para-orthogonal functions
(PORF) by

$\displaystyle
Q_{n}^{\nu}(z,\tau_{n}^{\nu})=\phi_{n}^{\nu}(z)+\tau_{n}^{\nu}\phi_{n}^{\nu*}(z),\qquad\tau_{n}^{\nu}\in\mathbb{T},\qquad\nu\in\\{\alpha,\beta,\gamma\\}.$
(6.1)

The PORF $Q_{n}^{\nu}(z,\tau_{n}^{\nu})$ is in $\mathcal{L}_{n}^{\nu}$ and it
is called para-orthogonal because it is not orthogonal to
$\mathcal{L}_{n-1}^{\nu}$ but it is orthogonal to a particular subspace of
$\mathcal{L}_{n-1}^{\nu}$ of dimension $n-1$.

######  Theorem 6.1.

The para-orthogonal rational function $Q_{n}(z)=Q_{n}(z,\tau_{n})$,
$\tau_{n}\in\mathbb{T}$ with $n\geq 2$ is orthogonal to
$\dot{\zeta}_{n}^{\alpha}\mathcal{L}_{n-1}\cap\dot{\zeta}_{n}^{\beta}\mathcal{L}_{n-1}=\zeta_{n}\mathcal{L}_{n-1}\cap\mathcal{L}_{n-1}=\mathcal{L}_{n-1}(\gamma_{n})$
with

$\displaystyle\mathcal{L}_{n-1}(\gamma_{n})=\\{f\in\mathcal{L}_{n-1}\colon
f(\gamma_{n})=0\\}=\left\\{\frac{\varpi_{n}^{*}(z)p_{n-2}(z)}{\pi_{n-1}(z)}\colon
p_{n-2}\in\mathcal{P}_{n-2}\right\\}.$

Recall that $\varpi_{n}^{*}(z)=z-\gamma_{n}$ if $\gamma_{n}\neq\infty$, and
$\varpi_{n}^{*}(z)=-1$ if $\gamma_{n}=\infty$.

###### Proof.

Suppose $\gamma_{n}=\alpha_{n}$ then
$\dot{\zeta}_{n}^{\alpha}=\zeta_{n}^{\alpha}$ and $\dot{\zeta}_{n}^{\beta}=1$.
Hence $\phi_{n}\perp\mathcal{L}_{n-1}$ and
$\phi_{n}^{*}\perp\zeta_{n}^{\alpha}\mathcal{L}_{n-1}$ and therefore
$Q_{n}\perp\mathcal{L}_{n-1}\cap\zeta_{n}^{\alpha}\mathcal{L}_{n-1}=\mathcal{L}_{n-1}(\alpha_{n})$.
The proof for $\gamma_{n}=\beta_{n}$ is similar. ∎

One may also define associated functions of the second kind as

$\displaystyle
P_{n}^{\nu}(z,\tau_{n})=\psi_{n}^{\nu}(z)-\tau_{n}^{\nu}\psi_{n}^{\nu*}(z),\qquad\nu\in\\{\alpha,\beta,\gamma\\}.$

From Theorems 3.9 and 5.6 we can easily obtain the following corollary.

######  Corollary 6.2.

With the notation $Q_{n}$ and $P_{n}$ for the PORF and the associated
functions just introduced, we have for $n\geq 1$

$\displaystyle
Q_{n}(z,\tau_{n})=\dot{B}_{n}^{\beta}(z)Q_{n}^{\alpha}(z,\tau_{n})=\tau_{n}\dot{B}_{n}^{\alpha}(z)Q_{n}^{\beta}(z,\overline{\tau}_{n})\qquad\mbox{if~{}~{}$\gamma_{n}=\alpha_{n}$},$

while

$\displaystyle
Q_{n}(z,\tau_{n})=\dot{B}_{n}^{\alpha}(z)Q_{n}^{\beta}(z,\tau_{n})=\tau_{n}\dot{B}_{n}^{\beta}(z)Q_{n}^{\alpha}(z,\overline{\tau}_{n})\qquad\mbox{if~{}~{}$\gamma_{n}=\beta_{n}$}.$

Similarly

$\displaystyle
P_{n}(z,\tau_{n})=\dot{B}_{n}^{\beta}(z)P_{n}^{\alpha}(z,\tau_{n})=-\tau_{n}\dot{B}_{n}^{\alpha}(z)P_{n}^{\beta}(z,\overline{\tau}_{n})\qquad\mbox{if~{}~{}$\gamma_{n}=\alpha_{n}$},$

while

$\displaystyle
P_{n}(z,\tau_{n})=\dot{B}_{n}^{\alpha}(z)P_{n}^{\beta}(z,\tau_{n})=-\tau_{n}\dot{B}_{n}^{\beta}(z)P_{n}^{\alpha}(z,\overline{\tau}_{n})\qquad\mbox{if~{}~{}$\gamma_{n}=\beta_{n}$}.$

###### Proof.

Assume that $\gamma_{n}=\alpha_{n}$, then
$\phi_{n}=\phi_{n}^{\alpha}\dot{B}_{n}^{\beta}$ and
$\phi_{n}^{*}=(\phi_{n}^{\alpha})^{*}\dot{B}_{n}^{\beta}$. Thus

$\displaystyle
Q_{n}(\cdot,\tau_{n})=\dot{B}_{n}^{\beta}\left[\phi_{n}^{\alpha}+\tau_{n}(\phi_{n}^{\alpha})^{*}\right]=\dot{B}_{n}^{\beta}Q_{n}^{\alpha}(\cdot,\tau_{n}).$

In a similar way one has

$\displaystyle
Q_{n}(\cdot,\tau_{n})=\phi_{n}^{\beta*}\dot{B}_{n}^{\alpha}+\tau_{n}\phi_{n}^{\beta}\dot{B}_{n}^{\alpha}=\tau_{n}\dot{B}_{n}^{\alpha}\left[\phi_{n}^{\beta}+\overline{\tau}_{n}\phi_{n}^{\beta*}\right]=\tau_{n}\dot{B}_{n}^{\alpha}Q_{n}^{\beta}(\cdot,\overline{\tau}_{n}).$

The proofs for $P_{n}$ and for $\gamma_{n}=\beta_{n}$ are similar. ∎

We are now ready to state that the zeros of the para-orthogonal rational
functions $Q_{n}^{\nu}(z,\tau_{n}^{\nu})$ will be simple and on $\mathbb{T}$
no matter whether $\nu=\alpha$, $\beta$ or $\gamma$.

######  Corollary 6.3.

Take $n\geq 1$, $\tau_{n}\in\mathbb{T}$ and $\nu\in\\{\alpha,\beta,\gamma\\}$
then the zeros of $Q_{n}^{\nu}(z,\tau_{n})$ are all simple and on
$\mathbb{T}$.

###### Proof.

The case $\nu=\alpha$ was proved in [9, Theorem 5.2.1] and by symmetry, this
also holds for $\nu=\beta$. For $\nu=\gamma$, assume that
$\gamma_{n}=\alpha_{n}$, then by the previous corollary
$Q_{n}(z,\tau_{n})=\dot{B}_{n}^{\beta}(z)Q_{n}^{\alpha}(z,\tau_{n})$ so that
the zeros of $Q_{n}(z,\tau_{n})$ are the zeros of $Q_{n}^{\alpha}(z,\tau_{n})$
(and hence will not cancel against any of the poles).

The proof for $\gamma_{n}=\beta_{n}$ is similar. ∎

These PORF properties are important because the zeros will deliver the nodes
for the rational Szegő quadrature as we will explain in the next section.

For further reference we give the following property.

######  Proposition 6.4.

For $n\geq 1$ and $\nu\in\\{\alpha,\beta,\gamma\\}$, the PORF satisfy, using
the notation of Theorem 5.1

$\displaystyle
Q_{n}^{\nu}(z,\tau_{n})=c_{n}^{\nu}\frac{\varpi_{n-1}^{\nu}(z)}{\varpi_{n}^{\nu}(z)}\big{[}\zeta_{n-1}^{\nu}(z)\phi_{n-1}^{\nu}(z)+\widetilde{\tau}_{n}^{\nu}\phi_{n-1}^{\nu*}(z)\big{]}$

with

$\displaystyle
c_{n}^{\nu}=e_{n}^{\nu}\big{(}\eta_{n1}+\eta_{n2}\tau_{n}\overline{\lambda}_{n}^{\nu}\big{)},\qquad\widetilde{\tau}_{n}^{\nu}=\frac{\widehat{\tau}_{n}+\lambda_{n}^{\nu}}{1+\widehat{\tau}_{n}\overline{\lambda}_{n}^{\nu}}\in\mathbb{T},\qquad\widehat{\tau}_{n}=\overline{\eta}_{n1}\eta_{n2}\tau_{n}.$

###### Proof.

Just take the recurrence relation of Theorem 5.1 and premultiply it with
$[1~{}~{}\tau_{n}]$. After re-arrangement of the terms, the expression above
will result. ∎

The importance of this property is that, up to a constant nonzero factor
$c_{n}^{\nu}$, we can compute $Q_{n}^{\nu}(z,\tau_{n})$ by exactly the same
recurrence that gives $\phi_{k}^{\nu}$ in terms of $\phi_{k-1}^{\nu}$ and
$\phi_{k-1}^{\nu*}$ for $k=1,2,\ldots,n$, except that we have to replace the
trailing parameter $\lambda_{n}^{\nu}$ by a unimodular constant
$\widetilde{\tau}_{n}^{\nu}$ to get $Q_{n}^{\nu}(z,\tau_{n})$. When we are
interested in the zeros of $Q_{n}^{\nu}(z,\tau_{n})$, then this normalizing
factor $c_{n}^{\nu}$ does not matter since it is nonzero.

##  7 Quadrature

We start by proving that the subspace of rational functions in which the
quadrature formulas will be exact only depends on the points
$\\{\alpha_{k}\colon k=0,\ldots,n-1\\}$ no matter whether the points
$\alpha_{k}$ are introduced as a pole $\alpha_{k}$ or
$\alpha_{k*}=1/\overline{\alpha}_{k}$ in the sequence
$\underline{\mbox{\boldmath{$\gamma$}}}_{n}=(\gamma_{k})_{k=0}^{n}$.

######  Lemma 7.1.

$\mathcal{R}_{n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathcal{L}_{n}\cdot\mathcal{L}_{n*}=\mathcal{R}_{n}^{\alpha}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathcal{L}_{n}^{\alpha}\cdot\mathcal{L}_{n*}^{\alpha}=\mathcal{R}_{n}^{\beta}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathcal{L}_{n}^{\beta}\cdot\mathcal{L}_{n*}^{\beta}$.

###### Proof.

The space $\mathcal{L}_{n}$ contains rational functions of degree at most $n$
whose denominators have zeros in $\\{\beta_{j}\colon
j\in\mathbbm{a}_{n}\\}\cup\\{\alpha_{j}\colon j\in\mathbbm{b}_{n}\\}$. The
space $\mathcal{L}_{n*}$ contains rational functions of degree at most $n$
whose denominators have zeros in $\\{\beta_{j}\colon
j\in\mathbbm{b}_{n}\\}\cup\\{\alpha_{j}\colon j\in\mathbbm{a}_{n}\\}$. Thus
the space $\mathcal{R}_{n}$ contains rational functions of degree at most $2n$
whose denominators have zeros in $\\{\beta_{j}\colon
j=1,\ldots,n\\}\cup\\{\alpha_{j}\colon j=1,\ldots,n\\}$. Since the
denominators of functions in $\mathcal{L}_{n}^{\alpha}$ have zeros in
$\\{\beta_{j}\colon j=1,\ldots,n\\}$ and functions in
$\mathcal{L}_{n*}^{\alpha}$ have zeros in $\\{\alpha_{j}\colon
j=1,\ldots,n\\}$, the denominator of functions in $\mathcal{R}_{n}^{\alpha}$
have zeros in $\\{\beta_{j}\colon j=1,\ldots,n\\}\cup\\{\alpha_{j}\colon
j=1,\ldots,n\\}$ so that $\mathcal{R}_{n}=\mathcal{R}_{n}^{\alpha}$. Of course
a similar argument shows that also $\mathcal{R}_{n}=\mathcal{R}_{n}^{\beta}$.
∎

As is well known from the classical case associated with the sequence
$\underline{\mbox{\boldmath{$\alpha$}}}$ discussed for example in [9, Chapter
5], the rational Szegő quadrature formulas are of the form

$\displaystyle
I_{n}^{\alpha}(f)=\sum_{j=0}^{n-1}w_{nj}^{\alpha}f(\xi_{nj}^{\alpha}),\qquad
n\geq 1,$

where $\xi_{nj}^{\alpha}=\xi_{nj}^{\alpha}(\tau_{n}^{\alpha})$,
$j=0,\ldots,n-1$ are the zeros of the para-orthogonal rational function
$Q_{n}^{\alpha}(z,\tau_{n}^{\alpha})$ with $\tau_{n}^{\alpha}\in\mathbb{T}$
and the weights are given by

$\displaystyle
w_{nj}^{\alpha}=w_{nj}^{\alpha}(\tau_{n})=\frac{1}{\sum\limits_{k=0}^{n-1}|\phi_{k}^{\alpha}(\xi_{nj}^{\alpha}(\tau_{n}^{\alpha}))|^{2}}>0.$

These formulas have maximal degree of exactness, meaning that

$\displaystyle\int_{\mathbb{T}}f(t)\mathrm{d}\mu(t)=I_{n}^{\alpha}(f),\qquad\forall\,f\in\mathcal{R}_{n-1}^{\alpha}$

and that no $n$-point interpolatory quadrature formula with nodes on
$\mathbb{T}$ and positive weights can be exact in a larger
$(2n-1)$-dimensional space of the form
$\mathcal{L}_{n}^{\alpha}\cdot\mathcal{L}_{(n-1)*}^{\alpha}$ or
$\mathcal{L}_{n*}^{\alpha}\cdot\mathcal{L}_{n-1}^{\alpha}$. They are however
exact in a subspace of dimension $2n-1$ of a different form as shown in [39]
and [17] for Laurent polynomials and in [11] for the rational case.

Our previous results show that taking an arbitrary order of selecting the
$\underline{\mbox{\boldmath{$\gamma$}}}$ does not add new quadrature formulas.
Indeed, if $\gamma_{n}=\alpha_{n}$ then the zeros of $Q_{n}(z,\tau_{n})$,
$Q_{n}^{\alpha}(z,\tau_{n})$ and $Q_{n}^{\beta}(z,\overline{\tau}_{n})$ are
all the same, i.e.,
$\xi_{nj}(\tau_{n})=\xi_{nj}^{\alpha}(\tau_{n})=\xi_{nj}^{\beta}(\overline{\tau}_{n})$,
$j=1,\ldots,n$. Dropping the dependency on $\tau$ from the notation, it is
directly seen that

$\displaystyle\sum_{k=0}^{n-1}|\phi_{k}(\xi_{nj})|^{2}=\sum_{k=0}^{n-1}|\phi_{k}^{\alpha}(\xi_{nj})\dot{B}_{k}^{\beta}(\xi_{nj})|^{2}=\sum_{k=0}^{n-1}|\phi_{k}^{\alpha}(\xi_{nj})|^{2}=\sum_{k=0}^{n-1}|\phi_{k}^{\alpha}(\xi_{nj}^{\alpha})|^{2}$

and thus we also have $w_{nj}=w_{nj}^{\alpha}$ and similarly
$w_{nj}=w_{nj}^{\beta}$. Therefore $I_{n}=I_{n}^{\alpha}=I_{n}^{\beta}$ for
appropriate choices of the defining parameters, i.e.,

$\displaystyle\tau_{n}^{\alpha}=\tau_{n}\qquad\mbox{and}\qquad\tau_{n}^{\beta}=\overline{\tau}_{n}\qquad\mbox{if~{}~{}$\gamma_{n}=\alpha_{n}$},$
$\displaystyle\tau_{n}^{\beta}=\tau_{n}\qquad\mbox{and}\qquad\tau_{n}^{\alpha}=\overline{\tau}_{n}\qquad\mbox{if~{}~{}$\gamma_{n}=\beta_{n}$}.$

An alternative expression for the weights is also known ([9, Theorem 5.4.1])
and it will obviously also coincide for $\nu\in\\{\alpha,\beta,\gamma\\}$:

$\displaystyle
w_{nj}^{\nu}=\frac{1}{2\xi_{nj}^{\nu}}\left.\frac{P_{n}^{\nu}(z)}{\frac{\mathrm{d}}{\mathrm{d}z}Q_{n}^{\nu}(z)}\right|_{z=\xi_{nj}^{\nu}},$

where $Q_{n}^{\nu}$ is the PORF with zeros $\\{\xi_{nj}^{\nu}\\}_{j=0}^{n-1}$
and $P_{n}^{\nu}$ the associated functions of the second kind as in Corollary
6.2. We have dropped again the obvious dependence on the parameter
$\tau_{n}^{\nu}$ from the notation.

A conclusion to be drawn from this section is that whether we choose the
sequence $\underline{\mbox{\boldmath{$\gamma$}}}$,
$\underline{\mbox{\boldmath{$\alpha$}}}$ or
$\underline{\mbox{\boldmath{$\beta$}}}$, the resulting quadrature formula we
obtain is an $n$-point formula that is exact in the space
$\mathcal{R}_{n-1}=\mathcal{L}_{n-1}\cdot\mathcal{L}_{(n-1)*}$. Such a
quadrature formula is called an $n$-point rational Szegő quadrature and these
are the only positive interpolating quadrature formulas exact in this space.
They are unique up to the choice of the parameter $\tau_{n}\in\mathbb{T}$.
Thus, to obtain the nodes and weights for a general sequence
$\underline{\mbox{\boldmath{$\gamma$}}}$ and a choice for
$\tau_{n}\in\mathbb{T}$, we can as well compute them for the sequence
$\underline{\mbox{\boldmath{$\alpha$}}}$ and an appropriate
$\tau_{n}^{\alpha}\in\mathbb{T}$ or for the sequence that alternates between
one $\alpha$ and one $\beta$. The resulting quadrature will be the same.

In practice nodes and weights of a quadrature formula are computed by solving
an eigenvalue problem. It will turn out that also in these computations,
nothing is gained from adopting a general order for the $\gamma$ sequence but
that a sequence alternating between $\alpha$ and $\beta$ (the balanced
situation) is the most economical one. We shall come back to this in Section
10 after we provide in the next sections the structure of the matrices whose
eigenvalues will give the nodes of the quadrature formula.

Rational interpolatory and Szegő quadrature formulas with arbitrary order of
the poles including error estimates and numerical experiments were considered
in [19].

##  8 Block diagonal with Hessenberg blocks

The orthogonal polynomials are a special case of ORF obtained by choosing a
sequence
$\underline{\mbox{\boldmath{$\gamma$}}}=\underline{\mbox{\boldmath{$\alpha$}}}$
that contains only zeros. Another special case is given by the orthogonal
Laurent polynomials (OLP) obtained by choosing an alternating sequence
$\underline{\mbox{\boldmath{$\gamma$}}}=(0,\infty,0,\infty,0,\infty,\ldots)$.
In [47], Velázquez described spectral methods for ORF on the unit circle that
were based on these two special choices. The result was that the shift
operator $\mathcal{T}_{\mu}\colon L_{\mu}^{2}\to L_{\mu}^{2}\colon f(z)\mapsto
zf(z)$ has a matrix representation that is a matrix Möbius transform of a
structured matrix. The structure is a Hessenberg structure in the case of
$\underline{\mbox{\boldmath{$\gamma$}}}=\underline{\mbox{\boldmath{$\alpha$}}}$
and it is a five-diagonal matrix (a so-called CMV matrix) for the sequence
$\underline{\mbox{\boldmath{$\gamma$}}}=(0,\beta_{1},\alpha_{2},\beta_{3},\alpha_{4},\ldots)$.
In the case of orthogonal (Laurent) polynomials the Möbius transform turns out
to be just the identity and we get the plain Hessenberg matrix for the
polynomials and the plain CMV matrix for the Laurent polynomials.

In [16], the OLP case was discussed using an alternative linear algebra
approach when the $0$, $\infty$ choice did not alternate nicely, but the order
in which they were added was arbitrary. Then the structured matrix generalized
to a so-called snake-shape matrix referring to the zig-zag shape of the
graphical representation that fixed the order in which the elementary factors
of the matrix had to be multiplied. This structure is a generalization of both
the Hessenberg and the CMV structures mentioned above. It is a block diagonal
where the blocks alternate between upper and lower Hessenberg structure.

To illustrate this for our ORF, we start in this section by using the approach
of Velázquez in [47], skipping all the details just to obtain the block
structure of the matrix that will represent the shift operator. In the next
sections we shall use the linear algebra approach of [16] to analyze the
computational aspects of obtaining the quadrature formulas.

We start from the recurrence for the $\phi_{k}^{\alpha}$ and transform it into
a recurrence relation for the $\phi_{k}$, which will eventually result in a
representation of the shift operator.

To avoid a tedious book-keeping of normalizing constants, we will just exploit
the fact that there are some constants such that certain dependencies hold.
For example the recurrence

$\displaystyle\phi_{n}^{\alpha}(z)=e_{n}\eta_{n1}\frac{\varpi_{n-1}^{\alpha}(z)}{\varpi_{n}^{\alpha}(z)}\big{[}\zeta_{n-1}^{\alpha}(z)\phi_{n-1}^{\alpha}(z)+\lambda_{n}^{\alpha}\phi_{n-1}^{\alpha*}(z)\big{]}$

or equivalently

$\displaystyle\phi_{n}^{\alpha}(z)=e_{n}\eta_{n1}\left[\sigma_{n-1}\frac{\varpi_{n-1}^{\alpha*}(z)}{\varpi_{n}^{\alpha}(z)}\phi_{n-1}^{\alpha}(z)+\lambda_{n}^{\alpha}\frac{\varpi_{n-1}^{\alpha}(z)}{\varpi_{n}^{\alpha}(z)}\phi_{n-1}^{\alpha*}(z)\right]$

or

$\displaystyle\varpi_{n-1}^{\alpha*}(z)\phi_{n-1}^{\alpha}(z)=e_{n}^{-1}\overline{\eta}_{n1}\overline{\sigma}_{n-1}\varpi_{n}^{\alpha}(z)\phi_{n}^{\alpha}(z)-\lambda_{n}^{\alpha}\overline{\sigma}_{n-1}\varpi_{n-1}^{\alpha}(z)\phi_{n-1}^{\alpha*}(z)$
(8.1)

will be expressed as

$\displaystyle\varpi_{n-1}^{\alpha*}\phi_{n-1}^{\alpha}\in\mathop{\rm
span}\nolimits\big{\\{}\varpi_{n}^{\alpha}\phi_{n}^{\alpha},\varpi_{n-1}^{\alpha}\phi_{n-1}^{\alpha*}\big{\\}}.$
(8.2)

Similarly, by combining the recurrence for $\phi_{n}^{\alpha}$ and
$\phi_{n}^{\alpha*}$ from Theorem 5.1 and eliminating $\phi_{n-1}^{\alpha}$,
we have

$\displaystyle\varpi_{n}^{\alpha}\phi_{n}^{\alpha*}\in\mathop{\rm
span}\nolimits\big{\\{}\varpi_{n}^{\alpha}\phi_{n}^{\alpha},\varpi_{n-1}^{\alpha}\phi_{n-1}^{\alpha*}\big{\\}}.$
(8.3)

Note that this relation always holds whether or not $\lambda_{n}^{\alpha}\neq
0$.

Now suppose that our sequence $\underline{\mbox{\boldmath{$\gamma$}}}$ has the
following configuration

$\displaystyle\underline{\mbox{\boldmath{$\gamma$}}}=(\ldots,\beta_{k-1},\alpha_{k},\ldots,\alpha_{n},\ldots,\alpha_{m-1},\beta_{m},\ldots,\beta_{l-1},\alpha_{l},\dots)$

with $2\leq k\leq n<m<l$ where $k$, $m$, $l$ are fixed and $n$ varying from
$k$ to $m-1$.

Using (8.2) and then repeatedly (8.3) we get

$\displaystyle\varpi_{n}^{\alpha*}\phi_{n}^{\alpha}\in\mathop{\rm
span}\nolimits\big{\\{}\varpi_{n+1}^{\alpha}\phi_{n+1}^{\alpha},\ldots,\varpi_{k}^{\alpha}\phi_{k}^{\alpha},\varpi_{k-1}^{\alpha}\phi_{k-1}^{\alpha*}\big{\\}}.$

Multiply this with
$\dot{B}_{n}^{\beta}=\dot{B}_{n-1}^{\beta}=\cdots=\dot{B}_{k-1}^{\beta}$

$\displaystyle\varpi_{n}^{\alpha*}\dot{B}_{n}^{\beta}\phi_{n}^{\alpha}\in\mathop{\rm
span}\nolimits\big{\\{}\varpi_{n+1}^{\alpha}\dot{B}_{n}^{\beta}\phi_{n+1}^{\alpha},\ldots,\varpi_{k}^{\alpha}\dot{B}_{k}^{\beta}\phi_{k}^{\alpha},\varpi_{k-1}^{\alpha}\dot{B}_{k-1}^{\beta}\phi_{k-1}^{\alpha*}\big{\\}},$

and since by Theorem 3.9, $\phi_{p}=\dot{B}_{p}^{\beta}\phi_{p}^{\alpha}$ for
$p=n,n-1,\ldots,k$ and $\phi_{k-1}=\dot{B}_{k-1}^{\beta}\phi_{k-1}^{\alpha*}$,
this becomes

$\displaystyle\varpi_{n}^{\alpha*}\phi_{n}\in\mathop{\rm
span}\nolimits\big{\\{}\varpi_{n+1}^{\alpha}\dot{B}_{n}^{\beta}\phi_{n+1}^{\alpha},\varpi_{n}^{\alpha}\phi_{n},\ldots,\varpi_{k}^{\alpha}\phi_{k},\varpi_{k-1}^{\alpha}\phi_{k-1}\big{\\}}.$
(8.4)

Thus if $n+1<m$, then $\gamma_{n+1}=\alpha_{n+1}$, hence
$\dot{B}_{n+1}^{\beta}=\dot{B}_{n}^{\beta}$ and
$\phi_{n+1}=\dot{B}_{n+1}^{\beta}\phi_{n+1}^{\alpha}$, so that

$\displaystyle\varpi_{n}^{\alpha*}\phi_{n}\in\mathop{\rm
span}\nolimits\\{\varpi_{p}^{\alpha}\phi_{p}\\}_{p=k-1}^{n+1},\qquad k\leq
n<m-1.$ (8.5)

If $n+1=m$, then $\gamma_{n+1}=\beta_{m}$ and we start dealing with the
subsequence of $\beta$’s in

$\displaystyle\underline{\mbox{\boldmath{$\gamma$}}}=(\ldots,\alpha_{m-1},\beta_{m},\ldots,\beta_{l-1},\alpha_{l},\ldots).$

Therefore we note that

$\displaystyle\varpi_{m}^{\alpha}(z)=(z-\alpha_{m})\frac{1-\overline{\alpha}_{m}z}{z-\alpha_{m}}=(z-\alpha_{m})\frac{z-\beta_{m}}{1-\overline{\beta}_{m}z}\sigma_{m}^{2}=\sigma_{m}\varpi_{m}^{\alpha*}(z)\zeta_{m}^{\beta}(z).$
(8.6)

Thus

$\displaystyle\varpi_{m}^{\alpha}\dot{B}_{m-1}^{\beta}\phi_{m}^{\alpha}=\sigma_{m}\varpi_{m}^{\alpha*}\dot{B}_{m}^{\beta}\phi_{m}^{\alpha}.$

Using (8.2) again repeatedly we then see that

$\displaystyle\varpi_{m}^{\alpha}\dot{B}_{m-1}^{\beta}\phi_{m}^{\alpha}\in\dot{B}_{m}^{\beta}\mathop{\rm
span}\nolimits\big{\\{}\varpi_{m}^{\alpha}\phi_{m}^{\alpha*},\varpi_{m+1}^{\alpha}\phi_{m+1}^{\alpha}\big{\\}}=\dot{B}_{m}^{\beta}\mathop{\rm
span}\nolimits\big{\\{}\varpi_{m}^{\alpha}\phi_{m}^{\alpha*},\varpi_{m+1}^{\alpha*}\zeta_{m+1}^{\beta}\phi_{m+1}^{\alpha}\big{\\}}$
$\displaystyle\hphantom{\varpi_{m}^{\alpha}\dot{B}_{m-1}^{\beta}\phi_{m}^{\alpha}}{}=\mathop{\rm
span}\nolimits\big{\\{}\varpi_{m}^{\alpha}\dot{B}_{m}^{\beta}\phi_{m}^{\alpha*},\varpi_{m+1}^{\alpha*}\dot{B}_{m+1}^{\beta}\phi_{m+1}^{\alpha}\big{\\}}$
$\displaystyle\hphantom{\varpi_{m}^{\alpha}\dot{B}_{m-1}^{\beta}\phi_{m}^{\alpha}}{}=\cdots$
$\displaystyle\hphantom{\varpi_{m}^{\alpha}\dot{B}_{m-1}^{\beta}\phi_{m}^{\alpha}}{}=\mathop{\rm
span}\nolimits\big{\\{}\varpi_{m}^{\alpha}\dot{B}_{m}^{\beta}\phi_{m}^{\alpha*},\ldots,\varpi_{l-1}^{\alpha}\dot{B}_{l-1}^{\beta}\phi_{l-1}^{\alpha*},\varpi_{l}^{\alpha}\dot{B}_{l}^{\beta}\phi_{l}^{\alpha}\big{\\}}$
$\displaystyle\hphantom{\varpi_{m}^{\alpha}\dot{B}_{m-1}^{\beta}\phi_{m}^{\alpha}}{}=\mathop{\rm
span}\nolimits\big{\\{}\varpi_{m}^{\alpha}\phi_{m},\ldots,\varpi_{l-1}^{\alpha}\phi_{l-1},\varpi_{l}^{\alpha}\phi_{l}\big{\\}},$

so that by plugging this into (8.4) with $n+1=m$ gives

$\displaystyle\varpi_{m-1}^{\alpha*}\phi_{m-1}\in\mathop{\rm
span}\nolimits\big{\\{}\varpi_{p}^{\alpha}\phi_{p}\big{\\}}_{p=k-1}^{l}.$
(8.7)

The two relations (8.4) and (8.7) tell us how we should express
$\varpi_{n}^{\alpha*}\phi_{n}$ in terms of the
$\\{\varpi_{k}^{\alpha}\phi_{k}\colon k\leq n+1\\}$ in the case that
$\gamma_{n}=\alpha_{n}$.

The next step is to express $\varpi_{n}^{\alpha*}\phi_{n}$ in terms of
$\\{\varpi_{p}^{\alpha}\phi_{p}\\}$, $p\geq n-1$ when $\gamma_{n}=\beta_{n}$.
This goes along the same lines. Let us consider the mirror situation

$\displaystyle\underline{\mbox{\boldmath{$\gamma$}}}=(\ldots,\alpha_{m-1},\beta_{m},\ldots,\beta_{n},\ldots,\beta_{l-1},\alpha_{l},\ldots,\alpha_{r-1},\beta_{r},\ldots)$

with $2\leq m\leq n<l<r$ where $m$, $l$, $r$ are again fixed and $n$ is
varying from $m$ to $l-1$.

We first use (8.6) for $m=n$ to write

$\displaystyle\varpi_{n}^{\alpha*}\phi_{n}=\varpi_{n}^{\alpha*}\dot{B}_{n}^{\beta}\phi_{n}^{\alpha*}=\overline{\sigma}_{n}\varpi_{n}^{\alpha}\dot{B}_{n-1}^{\beta}\phi_{n}^{\alpha*},$

which implies that we shall need expressions for
$\varpi_{n}^{\alpha}\phi_{n}^{\alpha*}$. We use repeatedly (8.3) in
combination with the previous relation to get

$\displaystyle\varpi_{n}^{\alpha*}\phi_{n}\in\dot{B}_{n-1}^{\beta}\mathop{\rm
span}\nolimits\big{\\{}\varpi_{n-1}^{\alpha}\phi_{n-1}^{\alpha*},\varpi_{n}^{\alpha}\phi_{n}^{\alpha}\big{\\}}$
$\displaystyle\hphantom{\varpi_{n}^{\alpha*}\phi_{n}}{}=\mathop{\rm
span}\nolimits\big{\\{}\dot{B}_{n-1}^{\beta}\varpi_{n-1}^{\alpha}\phi_{n-1}^{\alpha*},\dot{B}_{n}^{\beta}\varpi_{n}^{\alpha*}\phi_{n}^{\alpha}\big{\\}}$
$\displaystyle\hphantom{\varpi_{n}^{\alpha*}\phi_{n}}{}=\mathop{\rm
span}\nolimits\big{\\{}\dot{B}_{n-1}^{\beta}\varpi_{n-1}^{\alpha}\phi_{n-1}^{\alpha*},\dot{B}_{n}^{\beta}\varpi_{n}^{\alpha}\phi_{n}^{\alpha*},\dot{B}_{n}^{\beta}\varpi_{n+1}^{\alpha}\phi_{n+1}^{\alpha}\big{\\}}$
$\displaystyle\hphantom{\varpi_{n}^{\alpha*}\phi_{n}}{}=\mathop{\rm
span}\nolimits\big{\\{}\dot{B}_{n-1}^{\beta}\varpi_{n-1}^{\alpha}\phi_{n-1}^{\alpha*},\dot{B}_{n}^{\beta}\varpi_{n}^{\alpha}\phi_{n}^{\alpha*},\dot{B}_{n+1}^{\beta}\varpi_{n+1}^{\alpha*}\phi_{n+1}^{\alpha}\big{\\}}$
$\displaystyle\hphantom{\varpi_{n}^{\alpha*}\phi_{n}}{}=\mathop{\rm
span}\nolimits\big{\\{}\dot{B}_{n-1}^{\beta}\varpi_{n-1}^{\alpha}\phi_{n-1}^{\alpha*},\dot{B}_{n}^{\beta}\varpi_{n}^{\alpha}\phi_{n}^{\alpha*},\dot{B}_{n+1}^{\beta}\varpi_{n+1}^{\alpha}\phi_{n+1}^{\alpha*},\dot{B}_{n+1}^{\beta}\varpi_{n+2}^{\alpha}\phi_{n+2}^{\alpha}\big{\\}}$
$\displaystyle\hphantom{\varpi_{n}^{\alpha*}\phi_{n}}{}=\cdots$
$\displaystyle\hphantom{\varpi_{n}^{\alpha*}\phi_{n}}{}=\mathop{\rm
span}\nolimits\big{\\{}\dot{B}_{n-1}^{\beta}\varpi_{n-1}^{\alpha}\phi_{n-1}^{\alpha*},\ldots,\dot{B}_{m-1}^{\beta}\varpi_{l-1}^{\alpha}\phi_{l-1}^{\alpha*},\dot{B}_{m-1}^{\beta}\varpi_{l}^{\alpha}\phi_{l}^{\alpha}\big{\\}}$
$\displaystyle\hphantom{\varpi_{n}^{\alpha*}\phi_{n}}{}=\mathop{\rm
span}\nolimits\big{\\{}\dot{B}_{n-1}^{\beta}\varpi_{n-1}^{\alpha}\phi_{n-1}^{\alpha*},\varpi_{n}^{\alpha}\phi_{n},\ldots,\varpi_{l-1}^{\alpha}\phi_{l-1},\varpi_{l}^{\alpha}\phi_{l}\big{\\}},$

where in the last step we used $\gamma_{l}=\alpha_{l}$ and hence
$\dot{B}_{l-1}^{\beta}=\dot{B}_{l}^{\beta}$ and
$\dot{B}_{l-1}^{\beta}\phi_{l}^{\alpha}=\phi_{l}$. Thus if $n>m$, also
$\gamma_{n-1}=\beta_{n-1}$ so that also the first term is
$\varpi_{n-1}^{\alpha}\phi_{n-1}$ and we have found that

$\displaystyle\varpi_{n}^{\alpha*}\phi_{n}\in\mathop{\rm
span}\nolimits\big{\\{}\varpi_{p}^{\alpha}\phi_{p}^{\alpha}\big{\\}}_{p=n-1}^{l},\qquad
m<n\leq l-1.$ (8.8)

Now we are left with the remaining case $n=m$, i.e.,
$\gamma_{n-1}=\alpha_{n-1}$ while $\gamma_{n}=\beta_{n}$. This is the missing
link that should connect a $\beta$ block to the previous $\alpha$ block at the
boundary $\ldots,\alpha_{m-1},\beta_{m},\ldots$.

Using (8.3) repeatedly we get

$\displaystyle\varpi_{m-1}^{\alpha}\phi_{m-1}^{\alpha*}\in\mathop{\rm
span}\nolimits\big{\\{}\varpi_{m-1}^{\alpha}\phi_{m-1}^{\alpha},\varpi_{m-2}^{\alpha}\phi_{m-2}^{\alpha*}\big{\\}}$
$\displaystyle\hphantom{\varpi_{m-1}^{\alpha}\phi_{m-1}^{\alpha*}}{}=\mathop{\rm
span}\nolimits\big{\\{}\varpi_{m-1}^{\alpha}\phi_{m-1}^{\alpha},\varpi_{m-2}^{\alpha}\phi_{m-2}^{\alpha},\varpi_{m-2}^{\alpha}\phi_{m-2}^{\alpha*}\big{\\}}$
$\displaystyle\hphantom{\varpi_{m-1}^{\alpha}\phi_{m-1}^{\alpha*}}{}=\cdots$
$\displaystyle\hphantom{\varpi_{m-1}^{\alpha}\phi_{m-1}^{\alpha*}}{}=\mathop{\rm
span}\nolimits\big{\\{}\varpi_{m-1}^{\alpha}\phi_{m-1}^{\alpha},\varpi_{m-2}^{\alpha}\phi_{m-2}^{\alpha},\ldots,\varpi_{k}^{\alpha}\phi_{k}^{\alpha},\varpi_{k-1}^{\alpha}\phi_{k-1}^{\alpha*}\big{\\}}.$

After multiplying with
$\dot{B}_{k}^{\beta}=\dot{B}_{k+1}^{\beta}=\cdots=\dot{B}_{n-1}^{\beta}$ we
get

$\displaystyle\varpi_{m-1}^{\alpha}\phi_{m-1}^{\alpha*}\in\mathop{\rm
span}\nolimits\big{\\{}\varpi_{p}^{\alpha}\phi_{p}\big{\\}}_{p=k-1}^{n-1}.$

We plug this in our previous expression for $\varpi_{n}^{\alpha*}\phi_{n}$ and
we arrive at

$\displaystyle\varpi_{m}^{\alpha*}\phi_{m}\in\mathop{\rm
span}\nolimits\big{\\{}\varpi_{p}^{\alpha}\phi_{p}\big{\\}}_{p=k-1}^{l}.$
(8.9)

To summarize, (8.4) and (8.7) in the case $\gamma_{n}=\alpha_{n}$ and (8.8)
and (8.9) in the case $\gamma_{n}=\beta_{n}$ show short recurrences for the
$\phi_{p}$ that fully rely on the recursion for the $\alpha$-related
quantities: $\phi_{n}^{\alpha}$ and $\phi_{n}^{\alpha*}$. They use only
factors $\varpi_{p}^{\alpha}$ and $\varpi_{p}^{\alpha*}$ and in the recurrence
relations only the rational Szegő parameters $\lambda_{p}^{\alpha}$ for the
$\alpha$ sequence are used. The longest recursions are needed to switch from a
$\alpha$ to a $\beta$ block, which is for $n\in\\{m-1,m\\}$ as given in (8.7)
and (8.9) since these need all elements form both blocks.

This analysis should illustrate that as long as we are in a succession of
$\alpha$’s we are building up an upper Hessenberg block. At the instant the
$\alpha$ sequence switches to a $\beta$ sequence one starts building a lower
Hessenberg block, which switches back to upper Hessenberg when again
$\alpha$’s enter the $\gamma$ sequence etc. See Fig. 1. Of course if there are
only $\alpha$’s in the sequence, we end up with just an upper Hessenberg as in
the classical case. If we alternate between one $\alpha$ and one $\beta$’s we
get the so called CMV type matrix which is a five-diagonal matrix with a
characteristic block structure a given in Fig. 2.

Suppose we start with a set of $\alpha$’s, and set

$\displaystyle\mathcal{A}=\mathrm{diag}(\alpha_{0},\alpha_{1},\ldots)$

and let us recycle our earlier notations
$\varpi_{n}(z)=1-\overline{\alpha}_{n}z$ and $\varpi_{n}^{*}(z)=z-\alpha_{n}$
to now act on an arbitrary operator $\mathcal{T}$ by doing the transform for
the whole sequence $\underline{\mbox{\boldmath{$\alpha$}}}$ simultaneously as
follows:
$\varpi_{\mathcal{A}}(\mathcal{T})=\mathcal{I}-\mathcal{T}\mathcal{A}^{*}$ and
$\varpi_{\mathcal{A}}^{*}(\mathcal{T})=\mathcal{T}-\mathcal{A}$ where
$\mathcal{I}$ is the identity operator. Then (at least formally) we have

$\displaystyle[\phi_{0},\phi_{1},\ldots]\varpi_{\mathcal{A}}^{*}(z\mathcal{I})=[\phi_{0},\phi_{1},\ldots]\varpi_{\mathcal{A}}(z\mathcal{I})\widehat{\mathcal{G}}$
(8.10)

with $\widehat{\mathcal{G}}$ having the structure shown in Fig. 1.

$=$

Figure 1: Structure of the matrix $\widehat{\mathcal{G}}$ which is the same as
the structure of the matrix $\mathcal{G}_{\mathcal{A}}$.

In the special case that the $\alpha$’s and $\beta$’s alternate as in

$\displaystyle\underline{\mbox{\boldmath{$\gamma$}}}=(\alpha_{1},\beta_{2},\alpha_{3},\beta_{4},\alpha_{5},\ldots),$

we get the five-diagonal matrix as in [12] for polynomials and for ORF in
[47].

Figure 2: Structure of the matrix $\widehat{\mathcal{G}}_{m}$ when $\alpha$’s
and $\beta$’s alternate, which is the five-diagonal CMV matrix given in [12,
47].

Since for $k\geq 1$ each $\alpha_{2k-1}$-column is the last in an $\alpha$
block and each $\beta_{2k}$-column is the first in a $\beta$ block, we get a
succession of $4\times 2$ blocks that shift down by two rows as illustrated by
Fig. 2.

The particular role of the last column in an $\alpha$ block and the first
column in a $\beta$ block is due to the fact that we have chosen to derive
these relations for the $\phi_{k}$ from the recurrence for the
$\phi_{k}^{\alpha}$ leading to factors $\varpi_{\mathcal{A}}$ and
$\varpi_{\mathcal{A}}^{*}$ and the use of parameters $\lambda_{k}^{\alpha}$. A
symmetric derivation could have been given by starting from the
$\phi_{k}^{\beta}$ recursion, in which case the $\beta$ blocks will correspond
to upper Hessenberg blocks and the $\alpha$ blocks to lower Hessenberg blocks.
Then the longer recurrence would occur in the last column of the $\beta$ block
and the first column of the $\alpha$ block. The shape of the matrix would be
the transposed shape of the figures shown here. However while it is rather
simple to deal with $\alpha_{j}=0$ in the current derivation, it requires
dealing with more technical issues to write down the formulas with the
corresponding $\beta_{j}$ equal to $\infty$. Therefore we stick to the present
form.

We can define as in [47] an operator Möbius transform of an operator
$\mathcal{T}$ by

$\displaystyle{\zeta}_{\mathcal{A}}(\mathcal{T})=\eta_{\mathcal{A}}\varpi_{\mathcal{A}}(\mathcal{T})^{-1}\varpi^{*}_{\mathcal{A}}(\mathcal{T})\eta_{\mathcal{A}^{*}}^{-1},\qquad\varpi_{\mathcal{A}}(\mathcal{T})=\mathcal{I}-\mathcal{T}\mathcal{A}^{*},\qquad\varpi_{\mathcal{A}}^{*}(\mathcal{T})=\mathcal{T}-\mathcal{A},$

and its inverse

$\displaystyle\widetilde{\zeta}_{\mathcal{A}}(\mathcal{T})=\eta_{\mathcal{A}}^{-1}\widetilde{\varpi}_{\mathcal{A}}^{*}(\mathcal{T})\widetilde{\varpi}_{\mathcal{A}}(\mathcal{T})^{-1}\eta_{\mathcal{A}^{*}},\qquad\widetilde{\varpi}_{\mathcal{A}}(\mathcal{T})=\mathcal{I}+\mathcal{A}^{*}\mathcal{T},\qquad\widetilde{\varpi}_{\mathcal{A}}^{*}(\mathcal{T})=\mathcal{T}+\mathcal{A},$

where

$\displaystyle\eta_{\mathcal{A}}:=\varpi_{\mathcal{A}}(\mathcal{A})^{1/2}=\sqrt{\mathcal{I}-\mathcal{A}\mathcal{A}^{*}}=\sqrt{\mathcal{I}-\mathcal{A}^{*}\mathcal{A}}=\eta_{\mathcal{A}^{*}}=\varpi_{\mathcal{A}^{*}}(\mathcal{A}^{*})^{1/2}$
$\displaystyle\hphantom{\eta_{\mathcal{A}}}{}=\mathrm{diag}\big{(}1,\sqrt{1-|\alpha_{1}|^{2}},\sqrt{1-|\alpha_{2}|^{2}},\ldots\big{)}.$

Recall the meaning of the $(\ )^{*}$ notation for a (constant) matrix from
Remark 3.5. Note that $\widetilde{\varpi}_{\mathcal{A}}$ is like
$\varpi_{\mathcal{A}}$, except that the minus sign is changed into a plus sign
and $\mathcal{T}\mathcal{A}^{*}$ is changed into $\mathcal{A}^{*}\mathcal{T}$.
It should also be clear that the alternating case (i.e., the case of the CMV
representation) gives the smallest possible bandwidth, as was also reported in
[12].

Now it is not difficult to see that (8.10) is equivalent to

$\displaystyle[\phi_{0}(z),\phi_{1}(z),\phi_{2}(z),\ldots]\big{(}z\mathcal{I}-\widetilde{\zeta}_{\mathcal{A}}(\mathcal{G})\big{)}=0,\qquad\widehat{\mathcal{G}}=\eta_{\mathcal{A}}^{-1}{\mathcal{G}}\,\eta_{\mathcal{A}^{*}}.$

To see this, exactly the same argument as used in [47] to derive relation (19)
of that paper can be applied here. Instead we give a direct derivation. The
relation (8.10) can be written as

$\displaystyle
0=\Phi\big{[}\varpi_{\mathcal{A}}^{*}(z\mathcal{I})-\varpi_{\mathcal{A}}(z\mathcal{I})\widehat{\mathcal{G}}\big{]}=\Phi\big{[}(z\mathcal{I}-\mathcal{A})-(\mathcal{I}-z\mathcal{A}^{*})\widehat{\mathcal{G}}\big{]}=\Phi\big{[}z\big{(}\mathcal{I}+\mathcal{A}^{*}\widehat{\mathcal{G}}\big{)}-\big{(}\mathcal{A}+\widehat{\mathcal{G}}\big{)}\big{]}.$

Since
$\widetilde{\varpi}_{\mathcal{A}}(\widehat{\mathcal{G}})=(\mathcal{I}+\mathcal{A}^{*}\widehat{\mathcal{G}})$
is invertible, this is equivalent with

$\displaystyle
0=\Phi\big{[}z\mathcal{I}-\widetilde{\varpi}_{\mathcal{A}}^{*}\big{(}\widehat{\mathcal{G}}\big{)}\widetilde{\varpi}_{\mathcal{A}}\big{(}\widehat{\mathcal{G}}\big{)}^{-1}\big{]}=\Phi\big{[}z\mathcal{I}-\eta_{\mathcal{A}}^{-1}\widetilde{\varpi}_{\mathcal{A}}^{*}(\mathcal{G})\widetilde{\varpi}_{\mathcal{A}}(\mathcal{G})^{-1}\eta_{\mathcal{A}^{*}}\big{]}=\Phi\big{[}z\mathcal{I}-\widetilde{\zeta}_{\mathcal{A}}(\mathcal{G})\big{]},$

where we used the fact that the matrix $\eta_{A}=\eta_{\mathcal{A}^{*}}$ and
its inverse commute with any diagonal matrix.

The matrices $\mathcal{G}$ and $\widehat{\mathcal{G}}$ have the same shape,
but $\mathcal{G}$ is rescaled to make it isometric. So we see that on
$\mathcal{L}=\mathop{\rm span}\nolimits\\{\phi_{0},\phi_{1},\ldots\\}$ the
operator $z\mathcal{I}$ has a matrix representation
$\widetilde{\zeta}_{\mathcal{A}}(\mathcal{G})$ with respect to the
$\phi$-basis. We shall have a more careful discussion of this fact in the next
section.

##  9 Factorization of a general CMV matrix

It is well known (see [6, 47] in the rational case and [16] for the polynomial
case) that the Hessenberg matrix $\mathcal{G}=\mathcal{H}$ (obtained when
$\underline{\mbox{\boldmath{$\gamma$}}}=\underline{\mbox{\boldmath{$\alpha$}}}$)
can be written as an infinite product of elementary matrices, i.e.,

$\displaystyle\mathcal{G}=\mathcal{H}=G_{1}G_{2}G_{3}G_{4}\cdots,$

where

$\displaystyle G_{k}:=\left[\begin{matrix}I_{k-1}&0&0\\\
0&\widetilde{G}_{k}&0\\\
0&0&I_{\infty}\end{matrix}\right],\qquad\widetilde{G}_{k}:=\left[\begin{matrix}-\delta_{k}&\eta_{k}\\\
\eta_{k}&\overline{\delta_{k}}\end{matrix}\right],\qquad\forall\,k\geq 1,$
(9.1)

where $I_{k-1}$ and $I_{\infty}$ are the identity matrices of sizes $(k-1)$
and $\infty$ respectively, $I_{0}$ is the empty matrix,
$\delta_{k}=\lambda_{k}^{\alpha}$ is the $k$-th Szegő parameter and
$\eta_{k}:=\sqrt{1-|\delta_{k}|^{2}}$.

Also the CMV matrix $\mathcal{C}^{\varepsilon}$ associated with the
alternating sequence
$\underline{\mbox{\boldmath{$\gamma$}}}=\underline{\mbox{\boldmath{$\varepsilon$}}}$
where $\gamma_{2k-1}=\alpha_{2k-1}$ and $\gamma_{2k}=\beta_{2k}$, for all
$k\geq 1$ uses _the same_ elementary transforms, but they are multiplied in a
different order:

$\displaystyle\mathcal{G}=\mathcal{C}_{o}\mathcal{C}_{e}=\left(\cdots
G_{9}G_{7}G_{5}G_{3}G_{1}\right)\cdot\left(G_{2}G_{4}G_{6}G_{8}G_{10}\cdots\right)$
$\displaystyle\hphantom{G}{}=\left(G_{1}G_{3}G_{5}G_{7}G_{9}\cdots\right)\cdot\left(G_{2}G_{4}G_{6}G_{8}G_{10}\cdots\right).$

To find explicit expressions for these elementary factors in our case, taking
into account the proper normalization, requires a more detailed analysis. We
start with the following

######  Theorem 9.1.

In the case
$\underline{\mbox{\boldmath{$\gamma$}}}=\underline{\mbox{\boldmath{$\alpha$}}}$,
the recurrence relation of Theorem 5.1 can be rewritten in this form

$\displaystyle\left[\begin{matrix}\displaystyle\frac{\varpi_{n-1}^{\alpha*}\phi_{n-1}^{\alpha}}{\sqrt{1-|\alpha_{n-1}|^{2}}}&\displaystyle\frac{\varpi_{n}^{\alpha}\phi_{n}^{\alpha*}}{\sqrt{1-|\alpha_{n}|^{2}}}\end{matrix}\right]=\left[\begin{matrix}\displaystyle\frac{\varpi_{n-1}^{\alpha}\phi_{n-1}^{\alpha*}}{\sqrt{1-|\alpha_{n-1}|^{2}}}&\displaystyle\frac{\varpi_{n}^{\alpha}\phi_{n}^{\alpha}}{\sqrt{1-|\alpha_{n}|^{2}}}\end{matrix}\right]\widetilde{G}_{n}^{\alpha}$
(9.2)

with

$\displaystyle\widetilde{G}_{n}^{\alpha}=\overline{\sigma}_{n-1}\overline{\eta}^{\alpha}_{n1}\left[\begin{matrix}-\lambda_{n}^{\alpha}\eta_{n1}^{\alpha}&\sqrt{1-|\lambda_{n}^{\alpha}|^{2}}\\\
\sqrt{1-|\lambda_{n}^{\alpha}|^{2}}&\overline{\lambda_{n}^{\alpha}}\overline{\eta}^{\alpha}_{n1}\end{matrix}\right]\left[\begin{matrix}1&0\\\
0&\sigma_{n}\end{matrix}\right]$

or

$\displaystyle\left[\begin{matrix}\varpi_{n-1}^{\alpha*}\phi_{n-1}^{\alpha}&\varpi_{n}^{\alpha}\phi_{n}^{\alpha*}\end{matrix}\right]=\left[\begin{matrix}\varpi_{n-1}^{\alpha}\phi_{n-1}^{\alpha*}&\varpi_{n}^{\alpha}\phi_{n}^{\alpha}\end{matrix}\right]\widehat{G}_{n}^{\alpha}$
(9.3)

with

$\displaystyle\widehat{G}_{n}^{\alpha}=\left[\begin{matrix}\big{(}1-|\alpha_{n-1}|^{2}\big{)}^{-1/2}&0\\\
0&\big{(}1-|\alpha_{n}|^{2}\big{)}^{-1/2}\end{matrix}\right]\widetilde{G}_{n}^{\alpha}\left[\begin{matrix}\big{(}1-|\alpha_{n-1}|^{2}\big{)}^{1/2}&0\\\
0&\big{(}1-|\alpha_{n}|^{2}\big{)}^{1/2}\end{matrix}\right].$

###### Proof.

This is in fact an explicit form of what in the previous section was expressed
as (8.2) and (8.3).

Taking the first line of the recurrence relation from Theorem 5.1 and solving
for $\varpi^{\alpha*}_{n-1}\phi^{\alpha}_{n-1}$ we get the explicit from of
(8.1), i.e.,

$\displaystyle\varpi_{n-1}^{\alpha*}\phi_{n-1}^{\alpha}=(e_{n}^{\alpha})^{-1}\overline{\eta}^{\alpha}_{n1}\overline{\sigma}_{n-1}\varpi_{n}^{\alpha}\phi_{n}^{\alpha}-\lambda_{n}^{\alpha}\overline{\sigma}_{n-1}\varpi_{n-1}^{\alpha}\phi_{n-1}^{\alpha*}.$

Using
$e_{n}^{\alpha}=\frac{\sqrt{1-|\alpha_{n}|^{2}}}{\sqrt{1-|\alpha_{n-1}|^{2}}}\frac{1}{\sqrt{1-|\lambda_{n}^{\alpha}|^{2}}}$
and
$\eta_{n2}^{\alpha}=\overline{\eta}_{n1}^{\alpha}\overline{\sigma}_{n-1}\sigma_{n}$
this becomes

$\displaystyle\frac{\varpi_{n-1}^{\alpha*}\phi_{n-1}^{\alpha}}{\sqrt{1-|\alpha_{n-1}|^{2}}}=\overline{\eta}^{\alpha}_{n1}\overline{\sigma}_{n-1}\left[-\lambda_{n}^{\alpha}\eta^{\alpha}_{n1}\frac{\varpi_{n-1}^{\alpha}\phi_{n-1}^{\alpha*}}{\sqrt{1-|\alpha_{n-1}|^{2}}}+\sqrt{1-|\lambda_{n}^{\alpha}|^{2}}\frac{\varpi_{n}^{\alpha}\phi_{n}^{\alpha}}{\sqrt{1-|\alpha_{n}|^{2}}}\right].$
(9.4)

The second line of the recurrence relation in Theorem 5.1 is

$\displaystyle\varpi_{n}^{\alpha}\phi_{n}^{\alpha*}=e^{\alpha}_{n}{\eta}^{\alpha}_{n2}{\sigma}_{n-1}\overline{\lambda_{n}^{\alpha}}\varpi_{n-1}^{\alpha*}\phi_{n-1}^{\alpha}+e_{n}^{\alpha}\eta_{n2}^{\alpha}\varpi_{n-1}^{\alpha}\phi_{n-1}^{\alpha*}.$

Eliminate the term $\varpi_{n-1}^{\alpha*}\phi_{n-1}^{\alpha}$ with the first
line of the recurrence and after inserting the value of $e_{n}^{\alpha}$ from
(5.1) and rearranging, we get

$\displaystyle\frac{\varpi_{n}^{\alpha}\phi_{n}^{\alpha}}{\sqrt{1-|\alpha_{n}|^{2}}}=\overline{\eta}_{n1}^{\alpha}\overline{\sigma}_{n-1}\left[\frac{\varpi_{n-1}^{\alpha}\phi_{n-1}^{\alpha}}{\sqrt{1-|\alpha_{n-1}|^{2}}}\sqrt{1-|\lambda_{n}^{\alpha}|^{2}}\sigma_{n}+\frac{\varpi_{n}^{\alpha}\phi_{n}^{\alpha}}{\sqrt{1-|\alpha_{n}|^{2}}}\overline{\lambda_{n}^{\alpha}}\overline{\eta}_{n1}^{\alpha}\sigma_{n}\right].$

This is equivalent with the formula (9.2). Derivation of the other form (9.3)
is obvious. ∎

Note that $\widetilde{G}_{n}^{\alpha}$ is unitary but
$\widehat{G}_{n}^{\alpha}$ is not unless $|\lambda_{n}^{\alpha}|=1$ or
$\alpha_{n}=\alpha_{n-1}$.

Now suppose that $n\in\mathbbm{a}_{n}$, i.e., $\gamma_{n}=\alpha_{n}$ then
$\dot{B}_{n}^{\beta}=\dot{B}_{n-1}^{\beta}$ and hence

$\displaystyle\left[\begin{matrix}\varpi_{n-1}^{\alpha*}\phi_{n-1}^{\alpha}\dot{B}_{n-1}^{\beta}&\varpi_{n}^{\alpha}\phi_{n}^{\alpha*}\dot{B}_{n}^{\beta}\end{matrix}\right]=\left[\begin{matrix}\varpi_{n-1}^{\alpha}\phi_{n-1}^{\alpha*}\dot{B}_{n-1}^{\beta}&\varpi_{n}^{\alpha}\phi_{n}^{\alpha}\dot{B}_{n}^{\beta}\end{matrix}\right]\widehat{G}^{\alpha}_{n}.$
(9.5)

If $n\in\mathbbm{b}_{n}$, i.e., $\gamma_{n}=\beta_{n}$, then
$\dot{B}_{n}^{\beta}=\dot{B}_{n-1}^{\beta}\zeta_{n}^{\beta}=\overline{\sigma}_{n}\dot{B}_{n-1}^{\beta}\frac{\varpi_{n}^{\alpha}}{\varpi_{n}^{\alpha*}}$.
Define then

$\displaystyle\widetilde{G}_{n}^{\beta}=\left[\begin{matrix}\overline{\sigma}_{n}&0\\\
0&1\end{matrix}\right]\widetilde{G}_{n}^{\alpha}\left[\begin{matrix}\sigma_{n}&0\\\
0&1\end{matrix}\right]=\widetilde{S}_{n}^{*}\widetilde{G}_{n}^{\alpha}\widetilde{S}_{n},\qquad\widetilde{S}_{n}=\mathrm{diag}(\sigma_{n},1),$
(9.6)

and as for $\widehat{G}_{n}^{\alpha}$

$\displaystyle\widehat{G}_{n}^{\beta}=\left[\begin{matrix}\big{(}1-|\alpha_{n-1}|^{2}\big{)}^{-1/2}&0\\\
0&\big{(}1-|\alpha_{n}|^{2}\big{)}^{-1/2}\end{matrix}\right]\widetilde{G}_{n}^{\beta}\left[\begin{matrix}\big{(}1-|\alpha_{n-1}|^{2}\big{)}^{1/2}&0\\\
0&\big{(}1-|\alpha_{n}|^{2}\big{)}^{1/2}\end{matrix}\right]$
$\displaystyle\hphantom{\widehat{G}_{n}^{\beta}}{}=\left[\begin{matrix}\overline{\sigma}_{n}&0\\\
0&1\end{matrix}\right]\widehat{G}_{n}^{\alpha}\left[\begin{matrix}\sigma_{n}&0\\\
0&1\end{matrix}\right],$

so that

$\displaystyle\left[\begin{matrix}\varpi_{n-1}^{\alpha}\phi_{n-1}^{\alpha*}\dot{B}_{n-1}^{\beta}&\varpi_{n}^{\alpha*}\phi_{n}^{\alpha}\dot{B}_{n}^{\beta}\end{matrix}\right]\widehat{G}^{\beta}_{n}$
$\displaystyle\qquad{}=\left[\begin{matrix}\varpi_{n-1}^{\alpha}\phi_{n-1}^{\alpha*}\dot{B}_{n-1}^{\beta}&\varpi_{n}^{\alpha}\phi_{n}^{\alpha}\dot{B}_{n-1}^{\beta}\end{matrix}\right]\overline{\sigma}_{n}\widehat{G}_{n}^{\alpha}\left[\begin{matrix}\sigma_{n}&0\\\
0&1\end{matrix}\right]$
$\displaystyle\qquad{}=\left[\begin{matrix}\varpi_{n-1}^{\alpha*}\phi_{n-1}^{\alpha}\dot{B}_{n-1}^{\beta}&\varpi_{n}^{\alpha}\phi_{n}^{\alpha*}\dot{B}_{n-1}^{\beta}\end{matrix}\right]\left[\begin{matrix}1&0\\\
0&\overline{\sigma}_{n}\end{matrix}\right]$
$\displaystyle\qquad{}=\left[\begin{matrix}\varpi_{n-1}^{\alpha*}\phi_{n-1}^{\alpha}\dot{B}_{n-1}^{\beta}&\varpi_{n}^{\alpha*}\phi_{n}^{\alpha*}\dot{B}_{n}^{\beta}\end{matrix}\right].$

Thus

$\displaystyle\left[\begin{matrix}\varpi_{n-1}^{\alpha*}\phi_{n-1}^{\alpha}\dot{B}_{n-1}^{\beta}&\varpi_{n}^{\alpha*}\phi_{n}^{\alpha*}\dot{B}_{n}^{\beta}\end{matrix}\right]=\left[\begin{matrix}\varpi_{n-1}^{\alpha}\phi_{n-1}^{\alpha*}\dot{B}_{n-1}^{\beta}&\varpi_{n}^{\alpha*}\phi_{n}^{\alpha}\dot{B}_{n}^{\beta}\end{matrix}\right]\widehat{G}^{\beta}_{n}.$
(9.7)

We build the general CMV matrix as a product of $G$-factors. Set
$\alpha_{0}=0$ and suppose we consider for $1\leq n<k<m<l<\cdots\leq\infty$

$\displaystyle\alpha_{0},\alpha_{1},\ldots,\alpha_{n-1}|\beta_{n},\ldots,\beta_{k-1}|\alpha_{k},\ldots,\alpha_{m-1}|\beta_{m},\ldots,\beta_{\ell-1}|\alpha_{\ell},\ldots.$

We use vertical bars to separate the different blocks. These separations are
of course clear from the notation $\alpha$ and $\beta$, but since that will
not so obvious when we denote the corresponding functions in the vectors
below, we introduced them here and keep them at the corresponding positions in
the vectors that we are about to introduce now. That should improve
readability of the formulas.

We consider the vector $\Phi=[\phi_{0},\phi_{1},\phi_{2},\ldots]$ so that

$\displaystyle\Phi(\mathcal{I}-\mathcal{A}^{*}z)=\big{[}\varpi_{0}^{\alpha}\phi_{0},\ldots,\varpi_{n-1}^{\alpha}\phi_{n-1}\big{|}\varpi_{n}^{\alpha}\phi_{n},\ldots,\varpi_{k-1}^{\alpha}\phi_{k-1}\big{|}\varpi_{k}^{\alpha}\phi_{k},\ldots$
$\displaystyle\hphantom{\Phi(\mathcal{I}-\mathcal{A}^{*}z)=\big{[}}{}\ldots,\varpi_{m-1}^{\alpha}\phi_{m-1}\big{|}\varpi_{m}^{\alpha}\phi_{m},\ldots,\varpi_{\ell-1}^{\alpha}\phi_{\ell-1}\big{|}\varpi_{\ell}^{\alpha}\phi_{\ell},\ldots\big{]}$
$\displaystyle\hphantom{\Phi(\mathcal{I}-\mathcal{A}^{*}z)}{}=\big{[}\varpi_{0}^{\alpha}\phi^{\alpha}_{0},\ldots,\varpi_{n-1}^{\alpha}\phi_{n-1}^{\alpha}\dot{B}_{n-1}^{\beta}\big{|}\varpi_{n}^{\alpha}\phi_{n}^{\alpha*}\dot{B}_{n}^{\beta},\ldots,\varpi_{k-1}^{\alpha}\phi_{k-1}^{\alpha*}\dot{B}_{k-1}^{\beta}\big{|}\varpi_{k}^{\alpha}\phi_{k}^{\alpha}\dot{B}_{k}^{\beta},\ldots$
$\displaystyle\hphantom{\Phi(\mathcal{I}-\mathcal{A}^{*}z)=\big{[}}{}\ldots,\varpi_{m-1}^{\alpha}\phi_{m-1}^{\alpha}\dot{B}_{m-1}^{\beta}\big{|}\varpi_{m}^{\alpha}\phi_{m}^{\alpha*}\dot{B}_{m}^{\beta},\ldots,\varpi_{\ell-1}^{\alpha}\phi_{\ell-1}^{\alpha*}\dot{B}_{\ell-1}^{\beta}\big{|}\varpi_{\ell}^{\alpha}\phi_{\ell}^{\alpha}\dot{B}_{\ell}^{\beta},\ldots\big{]}.$

Define for $\nu\in\\{\alpha,\beta\\}$ (recall
$\eta_{\mathcal{A}}=\sqrt{\mathcal{I}-\mathcal{A}^{*}\mathcal{A}}$)

$\displaystyle G_{n}^{\nu}=\left[\begin{matrix}I_{n-1}&0&0\\\
0&\widetilde{G}_{n}^{\nu}&0\\\
0&0&I_{\infty}\end{matrix}\right]=\mathrm{diag}(I_{n-1},\widetilde{G}_{n}^{\nu},I_{\infty}),\qquad\text{and}\qquad\widehat{G}_{n}^{\nu}=\eta_{\mathcal{A}}^{-1}G_{n}^{\nu}\eta_{\mathcal{A}}.$
(9.8)

Apply successively the $G_{n}^{\nu}$ to the right on the $\Phi$ vector. Keep
an increasing order for the factors within an $\alpha$ block and a decreasing
order for the factors covering a $\beta$ block. Thus for the example given
above the order will be

$\displaystyle\underline{\mbox{\boldmath{$\gamma$}}}=(\alpha_{0},\alpha_{1},\ldots,\alpha_{n-1}\|\beta_{n},\ldots,\beta_{k-1}|\alpha_{k}\|\alpha_{k+1},\ldots,\alpha_{m-1}\|\beta_{m},\ldots,\beta_{\ell-1}|\alpha_{\ell}\|\alpha_{\ell+1},\ldots),$
$\displaystyle\widehat{\mathcal{G}}=\underbrace{(\widehat{G}_{1}^{\alpha}\widehat{G}_{2}^{\alpha}\cdots\widehat{G}_{n-1}^{\alpha})}_{\displaystyle\widehat{G}_{\alpha}^{1}}\underbrace{(\widehat{G}_{k}^{\alpha}\widehat{G}_{k-1}^{\beta}\cdots\widehat{G}_{n}^{\beta})}_{\displaystyle\widehat{G}_{\beta}^{1}}\underbrace{(\widehat{G}_{k+1}^{\alpha}\cdots\widehat{G}_{m-1}^{\alpha})}_{\displaystyle\widehat{G}_{\alpha}^{2}}\underbrace{(\widehat{G}_{\ell}^{\alpha}\widehat{G}_{\ell-1}^{\beta}\cdots\widehat{G}_{m}^{\beta})}_{\displaystyle\widehat{G}_{\beta}^{2}}(\widehat{G}_{\ell+1}^{\alpha}\cdots)$
(9.9)
$\displaystyle\hphantom{\widehat{\mathcal{G}}}{}=\eta_{\mathcal{A}}^{-1}\underbrace{\underbrace{({G}_{1}^{\alpha}{G}_{2}^{\alpha}\cdots{G}_{n-1}^{\alpha})}_{\displaystyle{G}_{\alpha}^{1}}\underbrace{({G}_{k}^{\alpha}{G}_{k-1}^{\beta}\cdots{G}_{n}^{\beta})}_{\displaystyle{G}_{\beta}^{1}}\underbrace{({G}_{k+1}^{\alpha}\cdots{G}_{m-1}^{\alpha})}_{\displaystyle{G}_{\alpha}^{2}}\underbrace{({G}_{\ell}^{\alpha}{G}_{\ell-1}^{\beta}\cdots{G}_{m}^{\beta})}_{\displaystyle{G}_{\beta}^{2}}({G}_{\ell+1}^{\alpha}\cdots)}\eta_{\mathcal{A}}$
$\displaystyle\hphantom{\widehat{\mathcal{G}}}{}=\eta_{\mathcal{A}}^{-1}\rule{182.09763pt}{0.0pt}\mathcal{G}\rule{182.09763pt}{0.0pt}\eta_{\mathcal{A}}$

with $\widehat{G}_{n}^{\nu}$ and $G_{n}^{\nu}$, $\nu\in\\{\alpha,\beta\\}$ as
in (9.8) and similarly

$\displaystyle\widehat{G}_{\nu}^{n}=\eta_{\mathcal{A}}^{-1}G_{\nu}^{n}\eta_{\mathcal{A}},\qquad\text{for
$n=1,2,\ldots$; \ \
$\nu\in\\{\alpha,\beta\\}$}\qquad\text{and}\qquad\widehat{\mathcal{G}}=\eta_{\mathcal{A}}^{-1}\mathcal{G}\eta_{\mathcal{A}}.$
(9.10)

Note that we have aligned these formulas properly so that a shift in the
blocks from the first to the second line in (9.9) is made clear. The rationale
is that in a $G_{\beta}^{n}$ group the indices are decreasing from left to
right and in a $G_{\alpha}^{n}$ group the indices increase from left to right.
This is what defines the shape of the result. We used the vertical bars to
separate $\alpha$ and $\beta$ blocks in the sequence
$\underline{\mbox{\boldmath{$\gamma$}}}$ as before. Now we use double vertical
bars to indicate what corresponds to the $G_{\alpha}^{n}$ and $G_{\beta}^{n}$
blocks on the third line. On the second line in (9.9) you may note that the
first $\alpha$ in an $\alpha$ block of the $\gamma$ sequence corresponds to a
$\widehat{G}^{\alpha}$ factor that migrates to the front of the product group
of previous $\beta$ block. This is due to the fact that we analysed the
$\gamma$ sequence using the parameters $\lambda_{k}^{\alpha}$ of the $\alpha$
recurrence (recall the definitions of the $\widehat{G}_{k}^{\alpha}$ and
$\widehat{G}_{k}^{\beta}$ from Theorem 9.1 and equation (9.6)).

To see the effect of applying all these elementary operations, we start with
the first block $G_{\alpha}^{1}$. Using (9.5) when multiplying
$\Phi(\mathcal{I}-\mathcal{A}^{*}z)$ with $\widehat{G}_{\alpha}^{1}$ gives
(note $\phi_{0}=\phi_{0}^{*}$)

$\displaystyle\big{[}\varpi_{0}^{\alpha*}\phi_{0},\ldots,\varpi_{n-2}^{\alpha*}\phi_{n-2},\varpi_{n-1}^{\alpha}\phi_{n-1}^{\alpha*}\dot{B}_{n-1}^{\beta}\big{|}\varpi_{n}^{\alpha}\phi_{n},\ldots\big{]}.$