# Modulated Bi-orthogonal Polynomials on the Unit Circle:
The $2j-k$ and $j-2k$ Systems

###### Abstract.

We construct the systems of bi-orthogonal polynomials on the unit circle where
the Toeplitz structure of the moment determinants is replaced by
$\det(w_{2j-k})_{0\leq j,k\leq N-1}$ and the corresponding Vandermonde modulus
squared is replaced by $\prod_{1\leq j<k\leq
N}(\zeta_{k}-\zeta_{j})(\zeta^{-2}_{k}-\zeta^{-2}_{j})$. This is the simplest
case of a general system of $pj-qk$ with $p,q$ co-prime integers. We derive
analogues of the structures well known in the Toeplitz case: third order
recurrence relations, determinantal and multiple-integral representations,
their reproducing kernel and Christoffel-Darboux sum, and associated
(Carathéodory) functions. We close by giving full explicit details for the
system defined by the simple weight $w(\zeta)=e^{\zeta}$, which is a
specialisation of a weight arising from averages of moments of derivatives of
characteristic polynomials over $\textnormal{USp}(2N)$, $\textnormal{SO}(2N)$
and $\textnormal{O}^{-}(2N)$.

Roozbeh Gharakhloo***Department of Mathematics, Colorado State University,
Fort Collins, CO 80521, USA, E-mail<EMAIL_ADDRESS>Nicholas S. Witte†††School of Mathematics and Statistics, Victoria University
of Wellington, Wellington, New Zealand e-mail<EMAIL_ADDRESS>

  * 2020 Mathematics Subject Classification. 42A80, 42A52, 47B35, 33C45, 39A06, 11M50.

  * Keywords. bi-orthogonal polynomials on the unit circle, Toeplitz matrices, linear difference equations, random matrix theory

###### Contents

  1. 1 Motivation
    1. 1.1 Outline
  2. 2 Definitions and Notations
    1. 2.1 LDU Decomposition of $2j-k$ and $j-2k$ moment matrices
  3. 3 Bordered determinant representations
    1. 3.1 Connection of $2j-k$ and $j-2k$ polynomials
    2. 3.2 Biorthogonal polynomials in terms of the master determinants
    3. 3.3 The reproducing kernel in terms of the master determinants
  4. 4 Recurrence relations
    1. 4.1 Equivalent Dodgson Condensation identities
    2. 4.2 Relationships of polynomial tails, recurrence coefficients and determinants
  5. 5 Multiple integral representations
    1. 5.1 Multiple integral formulae for the bi-orthogonal polynomials
    2. 5.2 Multiple integral formulae for the reproducing kernels
    3. 5.3 The Christoffel-Darboux Identity
  6. 6 Associated Functions
  7. 7 The symplectic and orthogonal example
  8. 8 Prospects of future work

##  1\. Motivation

The unitary group $U(N)$ with Haar (uniform) measure possesses the explicit
character formula of Weyl [71]

(1.1) $\frac{1}{(2\pi)^{N}N!}\prod_{1\leq j<k\leq
N}(\zeta_{k}-\zeta_{j})(\zeta^{-1}_{k}-\zeta^{-1}_{j}),\quad\zeta_{l}\coloneqq
e^{i\theta_{l}}\in{\mathbb{T}},\quad\theta_{l}\in(-\pi,\pi],$

where $\\{\zeta_{1},\dots,\zeta_{N}\\}\in{\rm Spec}(U)$ and
${\mathbb{T}}=\\{\zeta\in{\mathbb{C}}:|\zeta|=1\\}$. This also has an
interpretation as the joint probability density function (PDF) (see e.g. [30,
Chapter 2]) for the eigenphases of the Dyson circular ensemble (CUE). One
fundamental application of this formula is to characterise averages over $U\in
U(N)$ of class functions $c(U)$, i.e. symmetric functions of the eigenvalues
of $U$ only. An example of such functions are products
$\prod_{j=1}^{N}w(\zeta_{j})$ where $w(\zeta)$ may be interpreted as a weight
function or density. Introducing the Fourier components
$\\{w_{l}\\}_{l\in{\mathbb{Z}}}$ of this weight

(1.2) $w(\zeta)=\sum_{l=-\infty}^{\infty}w_{l}\zeta^{l},$

due to the well known Heine identity [70]

(1.3)
$\mathbb{E}_{U(N)}\big{[}\prod_{l=1}^{N}w(\zeta_{l})\big{]}=\det[w_{j-k}]_{j,k=0,\dots,N-1},$

we recognise that this is equivalent to studying Toeplitz determinants.
Intimately connected with the above problem are systems of bi-orthogonal
polynomials on the unit circle and its relationship to general, non-hermitian
(i.e. complex weight) Toeplitz determinants. The system of bi-orthogonal
polynomials $\\{\varphi_{n}(z),\bar{\varphi}_{n}(z)\\}^{\infty}_{n=0}$333The
bar notation in $\\{\bar{\varphi}_{n}(z)\\}^{\infty}_{n=0}$ is a standard
notation for the polynomials orthogonal to the polynomials
$\\{\varphi_{n}(z)\\}^{\infty}_{n=0}$ (see e.g. [35]). If the weight $w$ is
complex-valued the bar notation does not represent the complex conjugation,
while if the weight $w$ is real-valued the bar notation represents the complex
conjugation. with respect to the weight $w(\zeta)$ on the unit circle may be
defined by the orthogonality relation

(1.4) $\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta}{2\pi
i\zeta}w(\zeta)\varphi_{m}(\zeta)\bar{\varphi}_{n}(\bar{\zeta})=\delta_{m,n}.$

Such averages over the unitary group are ubiquitous in many areas of
mathematical physics, in particular the gap probabilities and characteristic
polynomial averages in the circular ensembles of random matrix theory
[2],[3],[34],[30], the spin-spin correlations of the planar Ising model
[63],[50], the density matrix of a system of impenetrable bosons on the ring
[31] and probability distributions for various classes of non-intersecting
lattice path problems [29].

Our study is motivated by yet another application of random matrix techniques,
in particular to analytic number theory. Random matrix models have been very
successful in constructing conjectures for estimating the integral moments of
central values in the $U(N)$ family of $L$-functions, through the works of
[51], [22], [23]. This program was extended by Ali̇ Altuğ et al [4] to the
three other families of $L$-functions: those characterised by the symmetry
types $\textnormal{USp}(2N)$, $\textnormal{SO}(2N)$ and
$\textnormal{O}^{-}(2N)$, and where the statistic of concern was the $n$-th
moment of the $m$-th derivative of the characteristic polynomial $\Lambda_{A}$
with $A\in\textnormal{USp}(2N),\textnormal{SO}(2N),\textnormal{O}^{-}(2N)$.
They computed

(1.5)
$M_{n}(\textnormal{G}(2N),m):=\int_{\textnormal{G}(2N)}\left(\Lambda^{(m)}_{A}(1)\right)^{n}\,\textrm{d}A,$

where G denotes USp, SO, or $\textnormal{O}^{-}$, and $\textrm{d}A$ is the
Haar measure on G, in the regime $N\to\infty$ and fixed $n,m$. Employing
similar techniques to [24] they found that the leading coefficient in the
large-$N$ expansion is proportional to the $n$-th derivative of
$e^{u}\mathcal{T}_{n,\ell}(u)$ where

(1.6) $\mathcal{T}_{n,\ell}(u):=\det\left(g_{2j-k+\ell}(u)\right)_{0\leq
j,k\leq n-1},$

for $n\geq 0$, $\ell\in{\mathbb{Z}}$ is fixed by the symmetry type and
$u\in{\mathbb{C}}$, and where

(1.7) $g_{l}(u)=\frac{1}{2\pi
i}\int_{{\mathbb{T}}}\frac{e^{\zeta+u\zeta^{-2}}}{\zeta^{l+1}}\,\textrm{d}\zeta=\frac{1}{\Gamma(l+1)}{}_{0}F_{2}\left(;\tfrac{1}{2}l+1,\tfrac{1}{2}(l+1);\tfrac{1}{4}u\right).$

Clearly for the $\textnormal{USp}(2N)$, $\textnormal{SO}(2N)$ and
$\textnormal{O}^{-}(2N)$ types the relevant moment determinant has the
structure

(1.8) $\det(w_{2j-k+r})_{0\leq j,k\leq n-1},$

and the corresponding joint density function has the form

(1.9) $\prod_{1\leq j<k\leq
n}\left(\zeta_{k}-\zeta_{j}\right)\left(\zeta^{-2}_{k}-\zeta^{-2}_{j}\right),\quad\zeta_{l}\coloneqq
e^{i\theta_{l}}\in{\mathbb{T}},\quad\theta_{l}\in(-\pi,\pi].$

Note that this joint density function is not real and positive, unlike the
Toeplitz case (1.1), and it will become clear that meaning can only be given
to more restrictive classes of weights than applies in the Toeplitz case, see
Theorems 3.1 and 3.2 for precise conditions on the existence of our systems.
For example the system with a constant weight or even a terminating Laurent
expansion of (1.2), i.e. a finite banded moment matrix, will not exist 444An
initial sequence of moment determinants can be non-zero, but the remainder
will vanish identically.. Of course one could take as the definition of a
putative system the modulus of (1.9) however this has some disadvantages and
we have chosen to pursue the analytic form here. If one takes the modulus of
(1.9) it factorises into three factors

(1.10) $\prod_{1\leq j<k\leq
n}\left|\zeta_{k}-\zeta_{j}\right|^{2}\left|\zeta_{k}+\zeta_{j}\right|,$

and one can see that this also applies to the joint density function

(1.11) $\prod_{1\leq j<k\leq
n}\left(\zeta^{2}_{k}-\zeta^{2}_{j}\right)\left(\zeta^{-1}_{k}-\zeta^{-1}_{j}\right),$

and thus any distinction is lost. More significantly is that the property of
complex analyticity would be lost in some aspects of the theory, and in
particular the differential structures whereby one requires spectral
derivatives $\textrm{d}/\textrm{d}\zeta$ of the bi-orthogonal polynomials and
their associated functions \- such derivatives form one member of key Lax
pairs of the integrable system. We will finish our discussion of these issues
with a final observation on the log-gas interpretation of Eq. (1.10): it has
the repulsion of coincident eigenvalues $\theta_{k}=\theta_{j},k\neq j$ with
strength $\beta=2$ but also a repulsion when $\theta_{k}=\theta_{j}\pm\pi$
with half this strength. It is our goal to extend the theory of Toeplitz
determinants and associated bi-orthogonal polynomial systems on the unit
circle to the case where they have a structure $pj-qk$ for two positive co-
prime integers, based upon (1.8) and (1.9). In the first example of such an
extension we consider the $p=2,q=1$ and $p=1,q=2$ cases exclusively, however
we will only employ techniques which are capable of generalisation to the
other cases.

Analogous systems of bi-orthogonal polynomials on the line with the Hankel
structure $\det(h_{pj+qk})_{0\leq j,k\leq N-1}$ have been studied for some
time - the area was initiated by Preiser [67]; general properties of these
systems were investigated by Konhauser [52], Ilyasov [46], Iserles and Nørsett
[47], [48]; explicit examples of cases related to Laguerre polynomials by
Konhauser [53], Carlitz [16], [17], Genin and Calvez [37], [38], Prabhakar
[66], Srivastava [69], Raizada [68]; and to the Jacobi polynomials by Madhekar
and Thakare [57], [59], [58], [60], [61]. As they stand the results reported
in the above works do not translate directly into the ones we seek. Systems of
bi-orthogonal Laurent polynomials on the line would be expected to provide an
equivalent framework to the system we treat here, however we prefer our
approach because of its direct linkage to the random matrix application
described earlier.

In the random matrix literature systems of bi-orthogonal polynomials on the
line are known as Muttalib-Borodin ensembles following the pioneering work of
[64] and [10]; Claeys and Romano have derived recurrence relations and a
scalar Riemann-Hilbert problem for general classes of weights [21]. See [33],
[32] for some selected recent developments. Other related lines of
investigation are the multiple orthogonal polynomial ensembles studied by
Kuijlaars and McLaughlin [55], and by Kuijlaars [56]. In the latter work
higher rank (i.e. greater than two) matrix systems of bi-orthogonal multiple
polynomials - the Nikishin systems and Angelesco systems - were discussed and
only the case of the Angelesco system (see Eq. (4.8)) with $p=2$ species of
$n_{1}=n_{2}=n$ particles $\\{x^{(1)}_{k}\\}_{k=1}^{n}$ ,
$\\{x^{(2)}_{k}\\}_{k=1}^{n}$ linked by
$x^{(2)}_{k}=-x^{(1)}_{k},k=1,\ldots,n$ and $w_{2}(x)=w_{1}(-x)$ has any
correspondence with our example \- it is actually proportional to the square
of (1.9).

It is also worth mentioning that on the Operator Theory side, the associated
operators on $L^{2}({\mathbb{T}})$ and their restrictions on the Hardy space
$H^{2}({\mathbb{T}})$ have been a subject of research in the last 25 years or
so. In a series of works [41, 42, 43, 44, 45], Mark C. Ho introduced and made
some fundamental studies on the $2j-k$ operators which he referred to as slant
Toeplitz operators. Later, Subhash C. Arora and Ruchika Batra studied the
properties of $pj-k$ operators ($p\in{\mathbb{N}}_{\geq 2}$) which they
referred to as generalized slant Toeplitz or p-th order slant Toeplitz
operators in a collection of papers [5, 6, 7, 8]. Particularly regarding the
large-size asymptotic analysis of the $pj-qk$ determinants, it is our hope
that the interplay between Operator Theory and the Riemann-Hilbert method
(which is rooted in the orthogonality structures studied this work and is the
subject of a future paper) could be made somewhat tractable. Examples of such
interplay for Toeplitz, bordered Toeplitz and Toeplitz+Hankel determinants can
be respectively found in [27], [9], and the introduction of [40].

###  1.1. Outline

Here we summarise the outline of our study. In §2 we present the definitions
of the $2j-k$ and $j-2k$ (master) determinants and the corresponding systems
of bi-orthogonal polynomials. In this section we also introduce the two main
tools for our analysis: the Dodgson Condensation identity and the multiple
integral formulae for the $2j-k$ and $j-2k$ determinants. The LDU
decompositions for the $2j-k$ and $j-2k$ determinants are also derived which
will be useful for proving the Christoffel-Darboux identity. In §3 we provide
the bordered determinant representations and prove the existence and
uniqueness of the systems of bi-orthogonal polynomials provided that the
associated determinants are non-zero. We also discuss the connection of $2j-k$
and $j-2k$ determinants and bi-orthogonal polynomials. Finally in this section
we express $2j-k$ and $j-2k$ bi-orthogonal polynomials and reproducing kernels
in terms of the corresponding master determinants. In §4 we derive pure-degree
and pure-offset recurrence relations for the $2j-k$ and $j-2k$ bi-orthogonal
polynomials. We also discuss equivalent Dodgson condensation identities which
result in the same recurrence relations and present several mixed recurrence
relations involving both $2j-k$ and $j-2k$ bi-orthogonal polynomials. In
subsection 4.2 we derive several relationships between polynomial tails,
recurrence coefficients and determinants of the $2j-k$ and $j-2k$ systems. In
§5 we prove multiple integral formulae for the $2j-k$ and $j-2k$ determinants,
bi-orthogonal polynomials, and reproducing kernels. We use some of these
formulae to derive representations of $Q$ and $R$-polynomials in terms of $P$
and $S$-polynomials, respectively. The multiple integral formulae are also
useful in deriving the Christoffel-Darboux identity in subsection 5.3. In §6
we introduce the associated functions and derive their multiple integral
formulae representations. We also find the corresponding Casorati matrices and
the first order recurrence relations they satisfy. In §7 we go back to the
weight relevant to the $L$-functions of the symmetry types
$\textnormal{USp}(2N)$, $\textnormal{SO}(2N)$ and $\textnormal{O}^{-}(2N)$. To
study a first concrete example, in this subsection we specifically study the
undeformed weight when $u=0$ where we find explicit formulae for the
determinants, Carathéodory functions, and the $2j-k$ polynomials. Eventually,
in §8, we will lay out a list of important open questions and the prospects of
future work. Throughout the paper, to highlight the distinguishing features of
these modulated bi-orthogonal systems and for the convenience of the reader we
try to make a comparison with the Toeplitz ($j-k$) theory whenever a result
about $2j-k$ and $j-2k$ systems is presented, mainly by referring to [35].

##  2\. Definitions and Notations

In this section we will define the objects studied in the paper and introduce
the necessary notations and conventions. Throughout the paper we respectively
use $j$ and $k$ as indices referring to the rows and the columns, and we
frequently use a boldfaced letter to distinguish the determinant of a matrix
with the matrix itself: $\det\bm{\mathscr{M}}\equiv\mathscr{M}$. Let
$\bm{\mathscr{M}}$ be an $n\times n$ matrix. By

(2.1) $\bm{\mathscr{M}}\left\\{\begin{matrix}j_{1}&j_{2}&\cdots&j_{\ell}\\\
k_{1}&k_{2}&\cdots&k_{\ell}\end{matrix}\right\\},\quad\text{and}\quad\mathscr{M}\left\\{\begin{matrix}j_{1}&j_{2}&\cdots&j_{\ell}\\\
k_{1}&k_{2}&\cdots&k_{\ell}\end{matrix}\right\\},$

we respectively mean the $(n-\ell)\times(n-\ell)$ matrix obtained from
$\bm{\mathscr{M}}$ by removing the rows $j_{i}$ and the columns $k_{i}$,
$1\leq i\leq\ell$, and its corresponding determinant. Although the order of
writing the row and column indices is immaterial for this definition, in this
work we prefer to respect the order of indices $j_{\ell_{1}}<j_{\ell_{2}}$ and
$k_{\ell_{1}}<k_{\ell_{2}}$ if $\ell_{1}<\ell_{2}$. We now recall the Dodgson
Condensation identity555which is also known as the Desnanot–Jacobi identity or
the Sylvester determinant identity. (see [1, 36, 14] and references therein)
which is an important tool for deriving many important relationships between
the objects studied in this work:

(2.2) $\mathscr{M}\cdot\mathscr{M}\left\\{\begin{matrix}j_{1}&j_{2}\\\
k_{1}&k_{2}\end{matrix}\right\\}=\mathscr{M}\left\\{\begin{matrix}j_{1}\\\
k_{1}\end{matrix}\right\\}\cdot\mathscr{M}\left\\{\begin{matrix}j_{2}\\\
k_{2}\end{matrix}\right\\}-\mathscr{M}\left\\{\begin{matrix}j_{1}\\\
k_{2}\end{matrix}\right\\}\cdot\mathscr{M}\left\\{\begin{matrix}j_{2}\\\
k_{1}\end{matrix}\right\\}.$

######  Definition 2.1.

For fixed offset values $r,s\in{\mathbb{Z}}$, we define the following
$(n+3)\times(n+3)$ master matrices of $2j-k$ and $j-2k$ structure:

(2.3) $\displaystyle\bm{\mathscr{D}}_{r}(z,\mathcal{z})$
$\displaystyle=\begin{pmatrix}w_{r}&w_{r-1}&\cdots&w_{r-n-1}&z^{n+1}\\\
w_{r+2}&w_{r+1}&\cdots&w_{r-n+1}&z^{n}\\\
\vdots&\vdots&\cdots&\vdots&\vdots\\\
w_{r+2n+2}&w_{r+2n+1}&\cdots&w_{r+n+1}&1\\\
1&\mathcal{z}&\cdots&\mathcal{z}^{n+1}&\star\\\ \end{pmatrix},$

(2.4) $\displaystyle\bm{\mathscr{E}}_{s}(z,\mathcal{z})$
$\displaystyle=\begin{pmatrix}w_{s}&w_{s-2}&\cdots&w_{s-2n-2}&z^{n+1}\\\
w_{s+1}&w_{s-1}&\cdots&w_{s-2n-1}&z^{n}\\\
\vdots&\vdots&\cdots&\vdots&\vdots\\\
w_{s+n+1}&w_{s+n-1}&\cdots&w_{s-n-1}&1\\\
1&\mathcal{z}&\cdots&\mathcal{z}^{n+1}&\star\\\ \end{pmatrix}.$

where

(2.5)
$w_{k}=\int_{{\mathbb{T}}}\zeta^{-k}w(\zeta)\frac{\textrm{d}\zeta}{2\pi\textrm{i}\zeta},$

is the $k$-th Fourier coefficient of the symbol
$w(z)=\sum_{\ell\in{\mathbb{Z}}}w_{l}z^{l}$.

In (2.3) and (2.4) for simplicity of notation we have suppressed the
dependence of $\bm{\mathscr{D}}_{r}(z,\mathcal{z})$ and
$\bm{\mathscr{E}}_{s}(z,\mathcal{z})$ on $n$. Also, throughout the paper, when
$z$ and $\mathcal{z}$ are not distinguished as distinct independent variables
we simply drop the arguments in the notation of the master matrices:

(2.6)
$\bm{\mathscr{D}}_{r}\equiv\bm{\mathscr{D}}_{r}(z,z)\quad\text{and}\quad\bm{\mathscr{E}}_{s}\equiv\bm{\mathscr{E}}_{s}(z,z).$

We use the determinants of master matrices $\bm{\mathscr{D}}_{r}$ and
$\bm{\mathscr{E}}_{s}$ to construct the $2j-k$ and $j-2k$ systems of bi-
orthogonal polynomials in §3, but since in all of those constructions either
the last row or the last column is removed, the entry $\star$ in (2.3) and
(2.4) never plays a role for construction of the bi-orthogonal polynomials and
thus is arbitrary for the purposes of this work. Let $\bm{D}^{(r)}_{n}$ and
$\bm{E}^{(s)}_{n}$ respectively denote the $n\times n$ matrices of $2j-k$ and
$j-2k$ structure and by $D^{(r)}_{n}$ and $E^{(s)}_{n}$ denote their
determinants:

(2.7) $D_{n}^{(r)}:=\det\begin{pmatrix}w_{r}&w_{r-1}&\cdots&w_{r-n+1}\\\
w_{r+2}&w_{r+1}&\cdots&w_{r-n+3}\\\ \vdots&\vdots&\vdots&\vdots\\\
w_{r+2n-2}&w_{r+2n-3}&\cdots&w_{r+n-1}\end{pmatrix}\equiv\underset{0\leq
j,k\leq n-1}{\det}\left(w_{r+2j-k}\right),$ (2.8)
$E_{n}^{(s)}:=\det\begin{pmatrix}w_{s}&w_{s-2}&\cdots&w_{s-2n+2}\\\
w_{s+1}&w_{s-1}&\cdots&w_{s-2n+3}\\\ \vdots&\vdots&\vdots&\vdots\\\
w_{s+n-1}&w_{s+n-3}&\cdots&w_{s-n+1}\end{pmatrix}\equiv\underset{0\leq j,k\leq
n-1}{\det}\left(w_{s+j-2k}\right).$

$D^{(r)}_{n}$ and $E^{(s)}_{n}$ can obviously be written in terms of the
determinant of master matrices $\bm{\mathscr{D}}_{r}$ and
$\bm{\mathscr{E}}_{s}$, as

(2.9) $D^{(r)}_{n}=\mathscr{D}_{r}\left\\{\begin{matrix}n&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\},\quad\text{and}\quad
E^{(s)}_{n}=\mathscr{E}_{s}\left\\{\begin{matrix}n&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\}.$

Also notice that

(2.10) $\displaystyle\bm{\mathscr{D}}_{r}\left\\{\begin{matrix}n&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\}$
$\displaystyle=\bm{\mathscr{D}}_{r}\left\\{\begin{matrix}0&n+1&n+2\\\
0&1&n+2\end{matrix}\right\\},$

(2.11) $\displaystyle\bm{\mathscr{E}}_{s}\left\\{\begin{matrix}n&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\}$
$\displaystyle=\bm{\mathscr{E}}_{s}\left\\{\begin{matrix}0&1&n+2\\\
0&n+1&n+2\end{matrix}\right\\}.$

######  Definition 2.2.

For an integrable function $f$ on the unit circle, we respectively define the
$2j-k$ and $j-2k$ multiple integrals as

(2.12)
$\mathcal{D}_{n}[f(\zeta)]:=\frac{1}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{2}}{2\pi\textrm{i}\zeta_{2}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}f(\zeta_{j})\prod_{1\leq
j<k\leq n}(\zeta_{k}-\zeta_{j})(\zeta^{-2}_{k}-\zeta^{-2}_{j}),$

and

(2.13)
$\mathcal{E}_{n}[f(\zeta)]:=\frac{1}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{2}}{2\pi\textrm{i}\zeta_{2}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}f(\zeta_{j})\prod_{1\leq
j<k\leq n}(\zeta^{2}_{k}-\zeta^{2}_{j})(\zeta^{-1}_{k}-\zeta^{-1}_{j}).$

In §5, in particular, we show the following multiple integral representations
for $D^{(r)}_{n}$ and $E^{(s)}_{n}$:

(2.14) $\displaystyle\mathcal{D}_{n}[w(\zeta)\zeta^{-r}]$
$\displaystyle=D^{(r)}_{n},$

(2.15) $\displaystyle\mathcal{E}_{n}[w(\zeta)\zeta^{-s}]$
$\displaystyle=E^{(s)}_{n}.$

######  Definition 2.3.

For each offset value $r\in{\mathbb{Z}}$, define the $2j-k$ sequences of monic
polynomials $\\{P_{n}(z;r)\\}^{\infty}_{n=0}$ and
$\\{Q_{n}(z;r)\\}^{\infty}_{n=0}$, $\deg P_{n}(z;r)=\deg Q_{n}(z;r)=n$,
satisfying the bi-orthogonality condition:

(2.16)
$\int_{{\mathbb{T}}}P_{m}(\zeta;r)Q_{n}(\zeta^{-2};r)\zeta^{-r}\frac{\textrm{d}\mu(\zeta)}{2\pi\textrm{i}\zeta}=h^{(r)}_{n}\delta_{mn},\qquad
m,n\in{\mathbb{N}}\cup\\{0\\},$

and for each offset value $s\in{\mathbb{Z}}$, define the $j-2k$ sequences of
monic polynomials $\\{R_{n}(z;s)\\}^{\infty}_{n=0}$ and
$\\{S_{n}(z;s)\\}^{\infty}_{n=0}$, $\deg R_{n}(z;s)=\deg S_{n}(z;s)=n$,
satisfying the bi-orthogonality condition:

(2.17)
$\int_{{\mathbb{T}}}R_{m}(\zeta^{2};s)S_{n}(\zeta^{-1};s)\zeta^{-s}\frac{\textrm{d}\mu(\zeta)}{2\pi\textrm{i}\zeta}=g^{(s)}_{n}\delta_{mn},\qquad
m,n\in{\mathbb{N}}\cup\\{0\\},$

where $h^{(r)}_{n}$ and $g^{(s)}_{n}$ are the norms of the polynomials squared
and $\textrm{d}\mu(\zeta)\equiv w(\zeta)\textrm{d}\zeta$ for some weight
function $w(z)$.

We will give representations of $h^{(r)}_{n}$ and $g^{(s)}_{n}$ as ratios of
$2j-k$ and $j-2k$ determinants in Theorems 3.1 and 3.2. Notice that the bi-
orthogonality condition (2.16) is equivalent to the orthogonality relations

(2.18)
$\int_{{\mathbb{T}}}P_{n}(\zeta;r)\zeta^{-2m-r}\frac{\textrm{d}\mu(\zeta)}{2\pi\textrm{i}\zeta}=h^{(r)}_{n}\delta_{mn},\qquad
m=0,1,\cdots,n,$

and

(2.19)
$\int_{{\mathbb{T}}}Q_{n}(\zeta^{-2};r)\zeta^{m-r}\frac{\textrm{d}\mu(\zeta)}{2\pi\textrm{i}\zeta}=h^{(r)}_{n}\delta_{mn},\qquad
m=0,1,\cdots,n.$

Similarly, the bi-orthogonality condition (2.17) is equivalent to the
orthogonality relations

(2.20)
$\int_{{\mathbb{T}}}R_{n}(\zeta^{2};s)\zeta^{-m-s}\frac{\textrm{d}\mu(\zeta)}{2\pi\textrm{i}\zeta}=g^{(s)}_{n}\delta_{mn},\qquad
m=0,1,\cdots,n,$

and

(2.21)
$\int_{{\mathbb{T}}}S_{n}(\zeta^{-1};s)\zeta^{2m-s}\frac{\textrm{d}\mu(\zeta)}{2\pi\textrm{i}\zeta}=g^{(s)}_{n}\delta_{mn},\qquad
m=0,1,\cdots,n.$

###  2.1. LDU Decomposition of $2j-k$ and $j-2k$ moment matrices

The linear space of polynomials $\\{P_{n}(z)\\}^{\infty}_{n=0}$,
$\\{Q_{n}(z)\\}^{\infty}_{n=0}$, etc. have expansions in an appropriate basis,
in this case the monomial basis $\\{z^{n}\\}^{\infty}_{n=-\infty}$ as befits a
Páde approximation problem with two fixed singularities $0,\infty$. We are
therefore led to consider the linear transformations from this preferred basis
to our orthogonal system. Let us denote the coefficients of $2j-k$ and $j-2k$
polynomials more precisely as

(2.22) $P_{n}(z;r)=\sum^{n}_{\ell=0}\mathcal{p}^{(r)}_{n,\ell}z^{\ell},\qquad
Q_{n}(z;r)=\sum^{n}_{\ell=0}\mathcal{q}^{(r)}_{n,\ell}z^{\ell},\qquad
R_{n}(z;s)=\sum^{n}_{\ell=0}\mathcal{r}^{(s)}_{n,\ell}z^{\ell},\qquad
S_{n}(z;s)=\sum^{n}_{\ell=0}\mathcal{s}^{(s)}_{n,\ell}z^{\ell}.$

with
$\mathcal{p}^{(r)}_{n,n}=\mathcal{q}^{(r)}_{n,n}=\mathcal{r}^{(s)}_{n,n}=\mathcal{s}^{(s)}_{n,n}=1$.
Let us also denote

(2.23) $\bm{Z}_{n}(z):=\begin{pmatrix}1\\\ z\\\ \vdots\\\
z^{n}\end{pmatrix}\quad\text{and}\quad\bm{F}_{n}(z;r):=\begin{pmatrix}F_{0}(z;r)\\\
F_{1}(z;r)\\\ \vdots\\\
F_{n}(z;r)\end{pmatrix},\qquad\bm{F}\in\\{\bm{P},\bm{Q},\bm{R},\bm{S}\\}.$

We thus have

(2.24)
$\bm{P}_{n}(z;r)=\bm{\mathcal{P}}_{n}^{(r)}\bm{Z}_{n}(z),\qquad\bm{Q}_{n}(z;r)=\bm{\mathcal{Q}}_{n}^{(r)}\bm{Z}_{n}(z),\qquad\bm{R}_{n}(z;s)=\bm{\mathcal{R}}_{n}^{(s)}\bm{Z}_{n}(z),\qquad\bm{S}_{n}(z;s)=\bm{\mathcal{S}}_{n}^{(s)}\bm{Z}_{n}(z),$

where
$\bm{\mathcal{P}}_{n}^{(r)},\bm{\mathcal{Q}}_{n}^{(r)},\bm{\mathcal{R}}_{n}^{(s)}$
and $\bm{\mathcal{S}}_{n}^{(s)}$ are the following $(n+1)\times(n+1)$ lower
triangular matrices

(2.25) $\bm{\mathcal{P}}_{n}^{(r)}:=\begin{pmatrix}1&0&\cdots&0\\\
\mathcal{p}^{(r)}_{1,0}&1&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
\mathcal{p}^{(r)}_{n,0}&\mathcal{p}^{(r)}_{n,1}&\cdots&1\end{pmatrix},\qquad\bm{\mathcal{Q}}_{n}^{(r)}:=\begin{pmatrix}1&0&\cdots&0\\\
\mathcal{q}^{(r)}_{1,0}&1&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
\mathcal{q}^{(r)}_{n,0}&\mathcal{q}^{(r)}_{n,1}&\cdots&1\end{pmatrix},$ (2.26)
$\bm{\mathcal{R}}_{n}^{(s)}:=\begin{pmatrix}1&0&\cdots&0\\\
\mathcal{\mathcal{r}}^{(s)}_{1,0}&1&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
\mathcal{r}^{(s)}_{n,0}&\mathcal{r}^{(s)}_{n,1}&\cdots&1\end{pmatrix},\qquad\bm{\mathcal{S}}_{n}^{(s)}:=\begin{pmatrix}1&0&\cdots&0\\\
\mathcal{s}^{(s)}_{1,0}&1&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
\mathcal{s}^{(s)}_{n,0}&\mathcal{s}^{(s)}_{n,1}&\cdots&1\end{pmatrix},$

whose inverses are also lower diagonal with $1$’s on the main diagonal. Also
let us denote the diagonal matrices of norms of polynomials by
$\bm{h}^{(r)}_{n}$ and $\bm{g}^{(r)}_{n}$:

(2.27) $\bm{h}^{(r)}_{n}:=\begin{pmatrix}h^{(r)}_{0}&\cdots&0\\\
\vdots&\ddots&\vdots\\\
0&\cdots&h^{(r)}_{n}\end{pmatrix},\quad\text{and}\quad\bm{g}^{(s)}_{n}:=\begin{pmatrix}g^{(s)}_{0}&\cdots&0\\\
\vdots&\ddots&\vdots\\\ 0&\cdots&g^{(s)}_{n}\end{pmatrix}.$

In the following result we give the LDU decompositions of the moment matrices
for the $2j-k$ and $j-2k$ systems which parallels the Toeplitz case closely,
see the unpublished work [62].

######  Theorem 2.1.

The LDU decompositions of $\bm{D}^{(r)}_{n+1}$ and $\bm{E}^{(s)}_{n+1}$ are
given by

(2.28) $\displaystyle\bm{D}^{(r)}_{n+1}$
$\displaystyle=\left[\bm{\mathcal{Q}}^{(r)}_{n}\right]^{-1}\bm{h}^{(r)}_{n}\left[\left(\bm{\mathcal{P}}^{(r)}_{n}\right)^{T}\right]^{-1},$

(2.29) $\displaystyle\bm{E}^{(s)}_{n+1}$
$\displaystyle=\left[\bm{\mathcal{S}}^{(s)}_{n}\right]^{-1}\bm{g}^{(s)}_{n}\left[\left(\bm{\mathcal{R}}^{(s)}_{n}\right)^{T}\right]^{-1}.$

###### Proof.

In this proof we drop the dependence of all functions and quantities on the
offsets $r$ and $s$ for simplicity of notation. We have

(2.30)
$\begin{split}h_{\nu}\delta_{\nu\mu}&=\int_{{\mathbb{T}}}P_{\nu}(\zeta)Q_{\mu}(\zeta^{-2})\zeta^{-r}w(\zeta)\frac{\textrm{d}\zeta}{2\pi\textrm{i}\zeta}=\sum_{m=0}^{\nu}\sum_{\ell=0}^{\mu}\mathcal{p}_{\nu,m}\mathcal{q}_{\mu,\ell}\int_{{\mathbb{T}}}\zeta^{m-2\ell-r}w(\zeta)\frac{\textrm{d}\zeta}{2\pi\textrm{i}\zeta}\\\
&=\sum_{m=0}^{\nu}\sum_{\ell=0}^{\mu}\mathcal{p}_{\nu,m}\mathcal{q}_{\mu,\ell}w_{2\ell-m+r}=\sum_{m=0}^{\nu}\sum_{\ell=0}^{\mu}\left(\bm{\mathcal{P}}_{n}\right)_{\nu,m}\left(\bm{\mathcal{Q}}_{n}\right)_{\mu,\ell}\left(\bm{D}_{n+1}\right)_{\ell,m}\\\
&=\sum_{m=0}^{\nu}\sum_{\ell=0}^{\mu}\left(\bm{\mathcal{Q}}_{n}\right)_{\mu,\ell}\left(\bm{D}_{n+1}\right)_{\ell,m}\left(\bm{\mathcal{P}}_{n}^{T}\right)_{m,\nu}=\left(\bm{\mathcal{Q}}_{n}\bm{D}_{n+1}\bm{\mathcal{P}}_{n}^{T}\right)_{\mu,\nu},\end{split}$

which is equivalent to (2.28). For the decomposition of $\bm{E}^{(s)}_{n+1}$
we consider

(2.31)
$\begin{split}g_{\nu}\delta_{\nu\mu}&=\int_{{\mathbb{T}}}R_{\nu}(\zeta^{2})S_{\mu}(\zeta^{-1})\zeta^{-s}w(\zeta)\frac{\textrm{d}\zeta}{2\pi\textrm{i}\zeta}=\sum_{m=0}^{\nu}\sum_{\ell=0}^{\mu}\mathcal{r}_{\nu,m}\mathcal{s}_{\mu,\ell}\int_{{\mathbb{T}}}\zeta^{2m-\ell-s}w(\zeta)\frac{\textrm{d}\zeta}{2\pi\textrm{i}\zeta}\\\
&=\sum_{m=0}^{\nu}\sum_{\ell=0}^{\mu}\mathcal{r}_{\nu,m}\mathcal{s}_{\mu,\ell}w_{\ell-2m+s}=\sum_{m=0}^{\nu}\sum_{\ell=0}^{\mu}\left(\bm{\mathcal{R}}_{n}\right)_{\nu,m}\left(\bm{\mathcal{S}}_{n}\right)_{\mu,\ell}\left(\bm{E}_{n+1}\right)_{\ell,m}\\\
&=\sum_{m=0}^{\nu}\sum_{\ell=0}^{\mu}\left(\bm{\mathcal{S}}_{n}\right)_{\mu,\ell}\left(\bm{E}_{n+1}\right)_{\ell,m}\left(\bm{\mathcal{R}}_{n}^{T}\right)_{m,\nu}=\left(\bm{\mathcal{S}}_{n}\bm{E}_{n+1}\bm{\mathcal{R}}_{n}^{T}\right)_{\mu,\nu},\end{split}$

which yields (2.29). ∎

##  3\. Bordered determinant representations

In this section we focus on determinantal representations of fundamental
elements in the theory where the moment matrix is bordered by rows or columns
containing basis vectors for the polynomial spaces. In the following result we
find that the bi-orthogonal polynomials of both systems can be represented as
bordered moment determinants in almost exactly the same way as for the
Toeplitz case, see e.g. Eq.(2.15,16) of [35] for comparison. As well as
revealing the conditions on the existence and uniqueness of these systems in a
simple way, this form will be very useful in our subsequent treatment.

######  Theorem 3.1.

If $D_{n}^{(r)}\neq 0$, the polynomials $P_{n}(z;r)$ and $Q_{n}(z;r)$ exist
and are uniquely given by

(3.1)
$P_{n}(z;r)=\frac{1}{D_{n}^{(r)}}\det\begin{pmatrix}w_{r}&w_{r-1}&\cdots&w_{r-n}\\\
w_{r+2}&w_{r+1}&\cdots&w_{r-n+2}\\\ \vdots&\vdots&\vdots&\vdots\\\
w_{r+2n-2}&w_{r+2n-3}&\cdots&w_{r+n-2}\\\ 1&z&\cdots&z^{n}\end{pmatrix},$

and

(3.2)
$Q_{n}(z;r)=\frac{1}{D_{n}^{(r)}}\det\begin{pmatrix}w_{r}&w_{r-1}&\cdots&w_{r-n+1}&1\\\
w_{r+2}&w_{r+1}&\cdots&w_{r-n+3}&z\\\ \vdots&\vdots&\vdots&\vdots&\vdots\\\
w_{r+2n}&w_{r+2n-1}&\cdots&w_{r+n+1}&z^{n}\end{pmatrix},$

from which one can observe that $h^{(s)}_{n}$ exists and can be written as

(3.3) $h^{(r)}_{n}=\frac{D_{n+1}^{(r)}}{D_{n}^{(r)}},\qquad
n\in{\mathbb{N}}\cup\\{0\\},\qquad D^{(r)}_{0}\equiv 1.$

Therefore if all $h^{(r)}_{\ell}$ exist and are non-zero for
$\ell=0,\cdots,n-1$, then

(3.4) $D^{(r)}_{n}=\prod_{\ell=0}^{n-1}h^{(r)}_{\ell}.$

###### Proof.

The existence simply follows from the fact that, if $D_{n}^{(r)}\neq 0$, we
can explicitly construct the system of monic bi-orthogonal polynomials
$P_{n}(z;r)$ and $Q_{n}(z;r)$ as in (3.1) and (3.2), respectively satisfying
(2.18) and (2.19). Now, let us discuss the uniqueness if $D_{n}^{(r)}\neq 0$.
Assume that
$P_{n}(z;r)=z^{n}+\displaystyle\sum^{n-1}_{\ell=0}\mathcal{p}^{(r)}_{n,\ell}z^{\ell}$
satisfies the orthogonality conditions (2.18). One can write the orthogonality
conditions (2.18) for $m=0,1,\cdots,n-1$ as the following linear system for
solving the constants $\mathcal{p}^{(r)}_{n,\ell}$, $0\leq\ell\leq n-1$:

(3.5) $\begin{pmatrix}w_{r}&w_{r-1}&\cdots&w_{r-n+1}\\\
w_{r+2}&w_{r+1}&\cdots&w_{r-n+3}\\\ \vdots&\vdots&\vdots&\vdots\\\
w_{r+2n-2}&w_{r+2n-3}&\cdots&w_{r+n-1}\end{pmatrix}\begin{pmatrix}\mathcal{p}^{(r)}_{n,0}\\\\[4.0pt]
\mathcal{p}^{(r)}_{n,1}\\\\[4.0pt] \vdots\\\\[4.0pt]
\mathcal{p}^{(r)}_{n,n-1}\end{pmatrix}=\begin{pmatrix}-w_{r-n}\\\
-w_{r-n+2}\\\ \vdots\\\ -w_{r+n-2}\end{pmatrix}.$

So, if $D_{n}^{(r)}\neq 0$, the above linear system has a unique solution, and
thus $P_{n}(z;r)$ is uniquely given by (3.1). Now, assume that
$Q_{n}(z;r)=z^{n}+\displaystyle\sum^{n-1}_{\ell=0}\mathcal{q}^{(r)}_{n,\ell}z^{\ell}$
satisfies the orthogonality conditions (2.19). We can write the orthogonality
conditions (2.19) for $m=0,1,\cdots,n-1$ as the following linear system for
solving the constants $\mathcal{q}^{(r)}_{n,\ell}$, $0\leq\ell\leq n-1$:

(3.6) $\begin{pmatrix}w_{r}&w_{r-1}&\cdots&w_{r-n+1}\\\
w_{r+2}&w_{r+1}&\cdots&w_{r-n+3}\\\ \vdots&\vdots&\vdots&\vdots\\\
w_{r+2n-2}&w_{r+2n-3}&\cdots&w_{r+n-1}\end{pmatrix}^{T}\begin{pmatrix}\mathcal{q}^{(r)}_{n,0}\\\
\mathcal{q}^{(r)}_{n,1}\\\ \vdots\\\
\mathcal{q}^{(r)}_{n,n-1}\end{pmatrix}=\begin{pmatrix}-w_{2n+r}\\\
-w_{2n+r-1}\\\ \vdots\\\ -w_{n+r+1}\end{pmatrix}.$

So, again, if $D_{n}^{(r)}\neq 0$, the above linear system can be inverted,
and thus has a unique solution. Therefore, $Q_{n}(z;r)$ is uniquely given by
(3.2). Finally, one can directly find (3.3) using (3.1) and (2.18), or
alternatively, using (3.2) and (2.19). ∎

In an identical manner, we can prove the following Theorem about existence and
uniqueness of $j-2k$ system of bi-orthogonal polynomials.

######  Theorem 3.2.

If $E_{n}^{(s)}\neq 0$, the polynomials $R_{n}(z;s)$ and $S_{n}(z;s)$ exist
and are uniquely given by

(3.7)
$R_{n}(z;s)=\frac{1}{E_{n}^{(s)}}\det\begin{pmatrix}w_{s}&w_{s-2}&\cdots&w_{s-2n}\\\
w_{s+1}&w_{s-1}&\cdots&w_{s-2n+1}\\\ \vdots&\vdots&\vdots&\vdots\\\
w_{s+n-1}&w_{s+n-3}&\cdots&w_{s-n-1}\\\ 1&z&\cdots&z^{n}\end{pmatrix},$

and

(3.8)
$S_{n}(z;s)=\frac{1}{E_{n}^{(s)}}\det\begin{pmatrix}w_{s}&w_{s-2}&\cdots&w_{s-2n+2}&1\\\
w_{s+1}&w_{s-1}&\cdots&w_{s-2n+3}&z\\\ \vdots&\vdots&\vdots&\vdots&\vdots\\\
w_{s+n}&w_{s+n-2}&\cdots&w_{s-n+2}&z^{n}\end{pmatrix},$

from which one can observe that $g^{(s)}_{n}$ exists and can be written as

(3.9) $g^{(s)}_{n}=\frac{E_{n+1}^{(s)}}{E_{n}^{(s)}},\qquad
n\in{\mathbb{N}}\cup\\{0\\},\qquad E^{(s)}_{0}\equiv 1.$

Therefore if all $g^{(s)}_{\ell}$ exist and are non-zero for
$\ell=0,\cdots,n-1$, then

(3.10) $E^{(s)}_{n}=\prod_{\ell=0}^{n-1}g^{(s)}_{\ell}.$

###  3.1. Connection of $2j-k$ and $j-2k$ polynomials

The $2j-k$ and $j-2k$ systems are intimately related by a duality through
exploiting the freedom to choose suitable offsets. This equivalence is only
exhibited in a formal algebraic rather than an analytical sense and reflects a
mapping of the interior of the unit circle to the exterior and vice-versa. In
the Toeplitz case one has a system of self-duality. Consequently the first of
such duality relations involves the determinants, which are related to one
another if one selects the appropriate offset values. More precisely, we have

(3.11) $D^{(r)}_{n}=E^{(r+n-1)}_{n},$

as these are reflections of each other across the main anti-diagonal. Indeed,
for any $n\times n$ matrix $\bm{\mathscr{M}}$ we have
$\det\bm{\mathscr{M}}^{\perp}=\det\bm{\mathscr{M}}$, which is due to the
identity
$\bm{\mathscr{M}}^{\perp}=\bm{\mathscr{A}}_{n}\bm{\mathscr{M}}^{T}\bm{\mathscr{A}}_{n},$
where $\bm{\mathscr{A}}_{n}$ is the $n\times n$ matrix with ones on the anti-
diagonal and zeros everywhere else, $\bm{\mathscr{M}}^{T}$ is the transpose of
the $n\times n$ matrix $\bm{\mathscr{M}}$, and $\bm{\mathscr{M}}^{\perp}$ is
the reflection of the $n\times n$ matrix $\bm{\mathscr{M}}$ across its main
anti-diagonal. Continuing the development of the duality theme our next result
furnishes further details in regard to the polynomials themselves. But first
let us recall a standard notation: for any polynomial $p$ of degree $n$, we
denote the reciprocal polynomial by $p^{*}$, that is

(3.12) $p^{*}(z):=z^{n}p(z^{-1}).$

######  Theorem 3.3.

The following identities hold between $2j-k$ and $j-2k$ polynomials

(3.13) $\displaystyle S^{*}_{n}(z;s)$
$\displaystyle=\frac{(-1)^{n}D^{(s-n+2)}_{n}}{E_{n}^{(s)}}P_{n}(z;s-n+2),$

(3.14) $\displaystyle R_{n}(z;s)$
$\displaystyle=\frac{(-1)^{n}D_{n}^{(s-n-1)}}{E_{n}^{(s)}}Q^{*}_{n}(z;s-n-1).$

###### Proof.

Computing $S^{*}_{n}(z;s)$ from (3.8) in view of (3.12) and reflection across
the anti-diagonal gives

(3.15)
$S^{*}_{n}(z;s):=\frac{1}{E_{n}^{(s)}}\det\begin{pmatrix}1&z&\cdots&z^{n}\\\
w_{s-n+2}&w_{s-n+1}&\cdots&w_{s-2n+2}\\\ \vdots&\vdots&\vdots&\vdots\\\
w_{s+n}&w_{s+n-1}&\cdots&w_{s}\end{pmatrix}.$

Now, we do $n$ consecutive adjacent row-swaps to move the first row to the
last which results in (3.13) in view of (3.1). The relationship (3.14) can be
found in an identical manner. ∎

###  3.2. Biorthogonal polynomials in terms of the master determinants

For many choices of triples $\\{a,b,c\\}\in{\mathbb{Z}}^{3}$, one can express
$z^{a}f_{n+b}(z;r+c)$, $f\in\\{P,Q^{*},R,S^{*}\\}$, in terms of the
determinants of master matrices (2.3) and (2.4). For each polynomial
$P,Q^{*},R$ and $S^{*}$, among the choices mentioned above, we select five
which turn out to be useful in connection to the specific Dodgson condensation
identities used in §4.

(3.16) $\displaystyle P_{n+1}(z;r)$
$\displaystyle=\frac{\mathscr{D}_{r}\left\\{\begin{matrix}n+1\\\
n+2\end{matrix}\right\\}}{\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}},$ (3.17) $\displaystyle P_{n}(z;r)$
$\displaystyle=\frac{\mathscr{D}_{r}\left\\{\begin{matrix}n&n+1\\\
n+1&n+2\end{matrix}\right\\}}{\mathscr{D}_{r}\left\\{\begin{matrix}n&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\}},$ (3.18) $\displaystyle P_{n}(z;r+2)$
$\displaystyle=\frac{\mathscr{D}_{r}\left\\{\begin{matrix}0&n+1\\\
n+1&n+2\end{matrix}\right\\}}{\mathscr{D}_{r+2}\left\\{\begin{matrix}n&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\}},$ (3.19) $\displaystyle zP_{n}(z;r-1)$
$\displaystyle=\frac{\mathscr{D}_{r}\left\\{\begin{matrix}n&n+1\\\
0&n+2\end{matrix}\right\\}}{\mathscr{D}_{r-1}\left\\{\begin{matrix}n&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\}},$

(3.20) $\displaystyle R_{n+1}(z;s)$
$\displaystyle=\frac{\mathscr{E}_{s}\left\\{\begin{matrix}n+1\\\
n+2\end{matrix}\right\\}}{\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}},$ (3.21) $\displaystyle R_{n}(z;s)$
$\displaystyle=\frac{\mathscr{E}_{s}\left\\{\begin{matrix}n&n+1\\\
n+1&n+2\end{matrix}\right\\}}{\mathscr{E}_{s}\left\\{\begin{matrix}n&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\}},$ (3.22) $\displaystyle R_{n}(z;s+1)$
$\displaystyle=\frac{\mathscr{E}_{s}\left\\{\begin{matrix}0&n+1\\\
n+1&n+2\end{matrix}\right\\}}{\mathscr{E}_{s+1}\left\\{\begin{matrix}n&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\}},$ (3.23) $\displaystyle zR_{n}(z;s-1)$
$\displaystyle=\frac{\mathscr{E}_{s}\left\\{\begin{matrix}0&n+1\\\
0&n+2\end{matrix}\right\\}}{\mathscr{E}_{s-1}\left\\{\begin{matrix}n&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\}},$

(3.24) $\displaystyle zP_{n}(z;r+1)$
$\displaystyle=\frac{\mathscr{D}_{r}\left\\{\begin{matrix}0&n+1\\\
0&n+2\end{matrix}\right\\}}{\mathscr{D}_{r+1}\left\\{\begin{matrix}n&n+1&n+2\\\
n&n+1&n+2\end{matrix},\right\\}},$

(3.25) $\displaystyle zR_{n}(z;s-2)$
$\displaystyle=\frac{\mathscr{E}_{s}\left\\{\begin{matrix}n&n+1\\\
0&n+2\end{matrix}\right\\}}{\mathscr{E}_{s-2}\left\\{\begin{matrix}n&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\}},$

(3.26) $\displaystyle Q^{*}_{n+1}(z;r)$
$\displaystyle=\frac{\mathscr{D}_{r}\left\\{\begin{matrix}n+2\\\
n+1\end{matrix}\right\\}}{\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}},$ (3.27) $\displaystyle Q^{*}_{n}(z;r+1)$
$\displaystyle=\frac{\mathscr{D}_{r}\left\\{\begin{matrix}0&n+2\\\
0&n+1\end{matrix}\right\\}}{\mathscr{D}_{r+1}\left\\{\begin{matrix}n&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\}},$ (3.28) $\displaystyle Q^{*}_{n}(z;r+2)$
$\displaystyle=\frac{\mathscr{D}_{r}\left\\{\begin{matrix}0&n+2\\\
n&n+1\end{matrix}\right\\}}{\mathscr{D}_{r+2}\left\\{\begin{matrix}n&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\}},$ (3.29) $\displaystyle zQ^{*}_{n}(z;r)$
$\displaystyle=\frac{\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
n&n+1\end{matrix}\right\\}}{\mathscr{D}_{r}\left\\{\begin{matrix}n&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\}},$ (3.30) $\displaystyle zQ^{*}_{n}(z;r-1)$
$\displaystyle=\frac{\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
0&n+1\end{matrix}\right\\}}{\mathscr{D}_{r-1}\left\\{\begin{matrix}n&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\}},$

(3.31) $\displaystyle S^{*}_{n+1}(z;s)$
$\displaystyle=\frac{\mathscr{E}_{s}\left\\{\begin{matrix}n+2\\\
n+1\end{matrix}\right\\}}{\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}},$ (3.32) $\displaystyle S^{*}_{n}(z;s-1)$
$\displaystyle=\frac{\mathscr{E}_{s}\left\\{\begin{matrix}0&n+2\\\
0&n+1\end{matrix}\right\\}}{\mathscr{E}_{s-1}\left\\{\begin{matrix}n&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\}},$ (3.33) $\displaystyle S^{*}_{n}(z;s+1)$
$\displaystyle=\frac{\mathscr{E}_{s}\left\\{\begin{matrix}0&n+2\\\
n&n+1\end{matrix}\right\\}}{\mathscr{E}_{s+1}\left\\{\begin{matrix}n&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\}},$ (3.34) $\displaystyle zS^{*}_{n}(z;s-2)$
$\displaystyle=\frac{\mathscr{E}_{r}\left\\{\begin{matrix}n+1&n+2\\\
0&n+1\end{matrix}\right\\}}{\mathscr{E}_{s-2}\left\\{\begin{matrix}n&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\}},$ (3.35) $\displaystyle zS^{*}_{n}(z;s)$
$\displaystyle=\frac{\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
n&n+1\end{matrix}\right\\}}{\mathscr{E}_{s}\left\\{\begin{matrix}n&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\}}.$

###  3.3. The reproducing kernel in terms of the master determinants

In the same spirit as the $j-k$ bi-orthogonal polynomials on the unit circle,
we define the reproducing kernels for the $2j-k$ and $j-2k$ systems.

######  Definition 3.1.

The reproducing kernel for the $2j-k$ and $j-2k$ systems respectively are
respectively defined as

(3.36) $\displaystyle K_{n}(z,\mathcal{z};r)$
$\displaystyle:=\sum_{j=0}^{n}\frac{1}{h^{(r)}_{j}}Q_{j}(z;r)P_{j}(\mathcal{z};r),$

(3.37) $\displaystyle L_{n}(z,\mathcal{z};s)$
$\displaystyle:=\sum_{j=0}^{n}\frac{1}{g^{(s)}_{j}}S_{j}(z;s)R_{j}(\mathcal{z};s).$

It is easy to see that $K_{n}(z,\mathcal{z};r)$ and $L_{n}(z,\mathcal{z};s)$
has the reproducing properties as described in the following equations:

(3.38)
$\int_{{\mathbb{T}}}K_{n}(z,\zeta;r)Q_{\ell}(\zeta^{-2};r)\zeta^{-r}w(\zeta)\frac{\textrm{d}\zeta}{2\pi\textrm{i}\zeta}=\begin{cases}Q_{\ell}(z;r),&0\leq\ell\leq
n,\\\ 0,&\ell>n,\end{cases}$ (3.39)
$\int_{{\mathbb{T}}}K_{n}(\zeta^{-2},\mathcal{z};r)P_{\ell}(\zeta;r)\zeta^{-r}w(\zeta)\frac{\textrm{d}\zeta}{2\pi\textrm{i}\zeta}=\begin{cases}P_{\ell}(\mathcal{z};r),&0\leq\ell\leq
n,\\\ 0,&\ell>n,\end{cases}$ (3.40)
$\int_{{\mathbb{T}}}L_{n}(z,\zeta^{2};s)S_{\ell}(\zeta^{-1};s)\zeta^{-s}w(\zeta)\frac{\textrm{d}\zeta}{2\pi\textrm{i}\zeta}=\begin{cases}S_{\ell}(z;s),&0\leq\ell\leq
n,\\\ 0,&\ell>n,\end{cases}$

and

(3.41)
$\int_{{\mathbb{T}}}L_{n}(\zeta^{-1},\mathcal{z};r)R_{\ell}(\zeta^{2};s)\zeta^{-s}w(\zeta)\frac{\textrm{d}\zeta}{2\pi\textrm{i}\zeta}=\begin{cases}R_{\ell}(\mathcal{z};s),&0\leq\ell\leq
n,\\\ 0,&\ell>n.\end{cases}$

Key properties flowing from these definitions are the normalisation relation

(3.42)
$\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta}{2\pi\textrm{i}\zeta}w(\zeta)\zeta^{-r}K_{n}(\zeta^{-2},\zeta;r)=n+1,$

and the projection relation

(3.43)
$\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta}{2\pi\textrm{i}\zeta}w(\zeta)\zeta^{-r}K_{n}(z_{1},\zeta;r)K_{n}(\zeta^{-2},z_{2};r)=K_{n}(z_{1},z_{2};r).$

Identical relations apply for the $L_{n}$ kernel with the obvious
modifications.

The above definitions of the reproducing kernels follows the pattern of the
Toeplitz case and the following result linking them to bordered moment
determinants reinforces this similarity. Although the Toeplitz case is not
quoted often it is almost identical after making the obvious modifications.

######  Theorem 3.4.

The reproducing kernels $K_{n}(z,\mathcal{z};r)$ and $L_{n}(z,\mathcal{z};s)$
can be represented in terms of the master determinants
$\mathscr{D}_{r}(z;\mathcal{z})$ and $\mathscr{E}_{s}(z;\mathcal{z})$ as

(3.44) $\displaystyle z^{n+1}K_{n+1}(z^{-1},\mathcal{z};r)$
$\displaystyle=-\frac{\mathscr{D}_{r}(z;\mathcal{z})}{D^{(r)}_{n+2}},$

(3.45) $\displaystyle z^{n+1}L_{n+1}(z^{-1},\mathcal{z};s)$
$\displaystyle=-\frac{\mathscr{E}_{s}(z;\mathcal{z})}{E^{(s)}_{n+2}}.$

###### Proof.

We only give the proof for (3.44), as (3.45) can be obtained in a similar way.
Define

(3.46)
$\widehat{K}_{n}(z,\mathcal{z};r):=-\frac{1}{D^{(r)}_{n+1}}\det\begin{pmatrix}w_{r}&w_{r-1}&\cdots&w_{r-n}&1\\\
w_{r+2}&w_{r+1}&\cdots&w_{r-n+2}&z\\\ \vdots&\vdots&\vdots&\vdots&\vdots\\\
w_{r+2n}&w_{r+2n-1}&\cdots&w_{r+n}&z^{n}\\\
1&\mathcal{z}&\cdots&\mathcal{z}^{n}&0\end{pmatrix}.$

One can easily establish that $\widehat{K}$ has the same reproducing
properties as $K$. To this end, its action on monomials is given by

(3.47)
$\int_{{\mathbb{T}}}\widehat{K}_{n}(z,\zeta;r)\zeta^{-2\ell}\zeta^{-r}w(\zeta)\frac{\textrm{d}\zeta}{2\pi\textrm{i}\zeta}=z^{\ell},\quad
0\leq\ell\leq n,$

and

(3.48)
$\int_{{\mathbb{T}}}\widehat{K}_{n}(\zeta^{-2},\mathcal{z};r)\zeta^{\ell}\zeta^{-r}w(\zeta)\frac{\textrm{d}\zeta}{2\pi\textrm{i}\zeta}=\mathcal{z}^{\ell},\quad
0\leq\ell\leq n.$

Therefore, we immediately get

(3.49)
$\int_{{\mathbb{T}}}\widehat{K}_{n}(z,\zeta;r)Q_{\ell}(\zeta^{-2};r)\zeta^{-r}w(\zeta)\frac{\textrm{d}\zeta}{2\pi\textrm{i}\zeta}=Q_{\ell}(z;r),\quad
0\leq\ell\leq n,$

and

(3.50)
$\int_{{\mathbb{T}}}\widehat{K}_{n}(\zeta^{-2},\mathcal{z};r)P_{\ell}(\zeta;r)\zeta^{-r}w(\zeta)\frac{\textrm{d}\zeta}{2\pi\textrm{i}\zeta}=P_{\ell}(\mathcal{z};r),\quad
0\leq\ell\leq n.$

Now we show that

(3.51) $\widehat{K}_{n}(z,\mathcal{z};r)=K_{n}(z,\mathcal{z};r),$

for all $z,\mathcal{z}\in{\mathbb{C}}$ and $r\in{\mathbb{Z}}$. Let us consider
the following $(n+2)\times(n+2)$ matrices

(3.52)
$\bm{\widehat{K}_{n}}(z,\zeta;r)=\begin{pmatrix}w_{r}&w_{r-1}&\cdots&w_{r-n}&1\\\
w_{r+2}&w_{r+1}&\cdots&w_{r-n+2}&z\\\ \vdots&\vdots&\vdots&\vdots&\vdots\\\
w_{r+2n}&w_{r+2n-1}&\cdots&w_{r+n}&z^{n}\\\
1&\zeta&\cdots&\zeta^{n}&0\end{pmatrix},$ (3.53)
$\bm{\widehat{\mathcal{P}}}^{(r)}_{n}:=\begin{pmatrix}1&0&\cdots&0&0\\\
p^{(r)}_{1,0}&1&\cdots&0&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
p^{(r)}_{n,0}&p^{(r)}_{n,1}&\cdots&1&0\\\
0&0&\cdots&0&1\end{pmatrix},\quad\text{and}\quad\bm{\widehat{\mathcal{Q}}}^{(r)}_{n}:=\begin{pmatrix}1&0&\cdots&0&0\\\
q^{(r)}_{1,0}&1&\cdots&0&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
q^{(r)}_{n,0}&q^{(r)}_{n,1}&\cdots&1&0\\\ 0&0&\cdots&0&1\end{pmatrix},$

Now observe that

(3.54)
$\begin{split}\bm{\widehat{\mathcal{Q}}}^{(r)}_{n}\bm{\widehat{K}_{n}}(z,\zeta;r)\left(\bm{\widehat{\mathcal{P}}}^{(r)}_{n}\right)^{T}&=\begin{pmatrix}\bm{\mathcal{Q}}^{(r)}_{n}&\bm{0}_{n+1}\\\
\bm{0}^{T}_{n+1}&1\end{pmatrix}\begin{pmatrix}\bm{D}_{n+1}&\bm{Z}_{n}(z)\\\
\bm{Z}^{T}_{n}(\zeta)&0\end{pmatrix}\begin{pmatrix}\left(\bm{\mathcal{P}}^{(r)}_{n}\right)^{T}&\bm{0}_{n+1}\\\
\bm{0}^{T}_{n+1}&1\end{pmatrix}\\\
&=\begin{pmatrix}\bm{\mathcal{Q}}^{(r)}_{n}\bm{D}_{n+1}\left(\bm{\mathcal{P}}^{(r)}_{n}\right)^{T}&\bm{\mathcal{Q}}^{(r)}_{n}\bm{Z}_{n}(z)\\\\[5.0pt]
\bm{Z}^{T}_{n}(\zeta)\left(\bm{\mathcal{P}}^{(r)}_{n}\right)^{T}&0\end{pmatrix}=\begin{pmatrix}\bm{h}^{(r)}_{n}&\bm{Q}_{n}(z;r)\\\
\bm{P}^{T}_{n}(\zeta;r)&0\end{pmatrix},\end{split}$

where the objects in the above equation are defined in subsection 2.1. Taking
the determinant of both sides of the above equation yields (3.51). Finally by
combining (3.51) and (3.46) and recalling (2.3) we obtain (3.44). ∎

##  4\. Recurrence relations

At this point we turn to the recurrence $n\mapsto n+1$ structures in the
$2j-k$ and $j-2k$ systems, focusing initially on the scalar forms. Throughout
this section we primarily use the Dodgson condensation identity (2.2) in order
to prove the following results, which furnish the tools that are capable of
extension to the more general $pj-qk$ systems.

Our first result yields extensions of the second order scalar difference
equations given in Eqs.(2.23,24) of [35] for the $2j-k$ system.

######  Theorem 4.1.

The third order pure-degree recurrence relations for the $2j-k$ polynomials
are given by

(4.1)
$P_{n+3}(z;r)-(\delta_{n+2}^{(r)}+\delta_{n+1}^{(r-1)})P_{n+2}(z;r)+(\delta_{n+1}^{(r-1)}\delta_{n+1}^{(r)}-z^{2})P_{n+1}(z;r)+(\delta_{n}^{(r)}+\eta^{(r-2)}_{n})z^{2}P_{n}(z;r)=0,$

and

(4.2)
$\begin{split}Q^{*}_{n+3}(z;r)&-(1+\beta^{(r)}_{n+2}z)Q^{*}_{n+2}(z;r)+(\beta^{(r)}_{n+1}+\alpha^{(r+1)}_{n+1}+\beta^{(r+1)}_{n+1}+\alpha^{(r+2)}_{n+1})zQ^{*}_{n+1}(z;r)\\\
&-(\beta^{(r+1)}_{n+1}+\alpha^{(r+2)}_{n+1})(\beta^{(r)}_{n}+\alpha^{(r+1)}_{n})z^{2}Q^{*}_{n}(z;r)=0,\end{split}$

where

(4.3) $\displaystyle\delta^{(r)}_{n}$
$\displaystyle=-\frac{h^{(r-1)}_{n}}{h^{(r)}_{n}},$ (4.4)
$\displaystyle\eta^{(r)}_{n}$
$\displaystyle=\frac{D_{n}^{(r+2)}D_{n+1}^{(r-1)}}{D^{(r)}_{n+1}D^{(r+1)}_{n}},$

(4.5) $\displaystyle\beta^{(r)}_{n}$
$\displaystyle=-\frac{h^{(r+2)}_{n}}{h^{(r)}_{n}},$ (4.6)
$\displaystyle\alpha^{(r)}_{n}$
$\displaystyle=\frac{D^{(r-1)}_{n}D^{(r+2)}_{n+1}}{D^{(r)}_{n+1}D^{(r+1)}_{n}}.$

###### Proof.

Let us consider the following Dodgson condensation identities (see (2.2)):

(4.7) $\begin{split}&\mathscr{D}_{r}\left\\{\begin{matrix}n+1\\\
n+2\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}n&n+1&n+2\\\
0&n+1&n+2\end{matrix}\right\\}=\mathscr{D}_{r}\left\\{\begin{matrix}n&n+1\\\
0&n+2\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}-\mathscr{D}_{r}\left\\{\begin{matrix}n&n+1\\\
n+1&n+2\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
0&n+2\end{matrix}\right\\},\end{split}$

(4.8) $\begin{split}&\mathscr{D}_{r}\left\\{\begin{matrix}n+1\\\
n+2\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}0&n+1&n+2\\\
0&n+1&n+2\end{matrix}\right\\}=\mathscr{D}_{r}\left\\{\begin{matrix}0&n+1\\\
0&n+2\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}-\mathscr{D}_{r}\left\\{\begin{matrix}0&n+1\\\
n+1&n+2\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
0&n+2\end{matrix}\right\\}.\end{split}$

The identity (4.7) can be written as

(4.9) $\frac{\mathscr{D}_{r}\left\\{\begin{matrix}n+1\\\
n+2\end{matrix}\right\\}}{\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}n&n+1&n+2\\\
0&n+1&n+2\end{matrix}\right\\}=\mathscr{D}_{r}\left\\{\begin{matrix}n&n+1\\\
0&n+2\end{matrix}\right\\}-\frac{\mathscr{D}_{r}\left\\{\begin{matrix}n&n+1\\\
n+1&n+2\end{matrix}\right\\}}{\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
0&n+2\end{matrix}\right\\}.$

So from (3.16), (3.17), (3.19), and (2.9) we have

(4.10)
$D^{(r-1)}_{n}P_{n+1}(z;r)=D^{(r-1)}_{n}zP_{n}(z;r-1)-\frac{D_{n}^{(r)}D_{n+1}^{(r-1)}}{D_{n+1}^{(r)}}P_{n}(z;r).$

Therefore from (3.3) we have

(4.11)
$P_{n+1}(z;r)=zP_{n}(z;r-1)+\delta^{(r)}_{n}P_{n}(z;r),\qquad\delta^{(r)}_{n}=-\frac{h^{(r-1)}_{n}}{h^{(r)}_{n}}.$

Let us also write (4.8) as

(4.12) $\frac{\mathscr{D}_{r}\left\\{\begin{matrix}n+1\\\
n+2\end{matrix}\right\\}}{\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}0&n+1&n+2\\\
0&n+1&n+2\end{matrix}\right\\}=\mathscr{D}_{r}\left\\{\begin{matrix}0&n+1\\\
0&n+2\end{matrix}\right\\}-\frac{\mathscr{D}_{r}\left\\{\begin{matrix}0&n+1\\\
n+1&n+2\end{matrix}\right\\}}{\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
0&n+2\end{matrix}\right\\}.$

Therefore from (3.16), (3.18), (3.24), and (2.9) we arrive at

(4.13)
$D^{(r+1)}_{n}P_{n+1}(z;r)=D^{(r+1)}_{n}zP_{n}(z;r+1)-\frac{D_{n}^{(r+2)}D_{n+1}^{(r-1)}}{D^{(r)}_{n+1}}P_{n}(z;r+2),$

and thus

(4.14)
$P_{n+1}(z;r)=zP_{n}(z;r+1)-\eta^{(r)}_{n}P_{n}(z;r+2),\qquad\eta^{(r)}_{n}=\frac{D_{n}^{(r+2)}D_{n+1}^{(r-1)}}{D^{(r)}_{n+1}D^{(r+1)}_{n}}.$

From (4.11) we have

(4.15) $zP_{n}(z;r-1)=P_{n+1}(z;r)-\delta_{n}^{(r)}P_{n}(z;r).$

From this, we immediately get

(4.16) $zP_{n}(z;r-2)=P_{n+1}(z;r-1)-\delta_{n}^{(r-1)}P_{n}(z;r-1).$

Using (4.15) again we get

(4.17)
$zP_{n}(z;r-2)=z^{-1}[P_{n+2}(z;r)-\delta_{n+1}^{(r)}P_{n+1}(z;r)]-\delta_{n}^{(r-1)}z^{-1}[P_{n+1}(z;r)-\delta_{n}^{(r)}P_{n}(z;r)],$

and therefore

(4.18)
$zP_{n}(z;r-2)=z^{-1}P_{n+2}(z;r)-z^{-1}(\delta_{n+1}^{(r)}+\delta_{n}^{(r-1)})P_{n+1}(z;r)+\delta_{n}^{(r-1)}\delta_{n}^{(r)}z^{-1}P_{n}(z;r).$

On the other hand, from (4.14) we have

(4.19) $P_{n+1}(z;r-2)=zP_{n}(z;r-1)-\eta^{(r-2)}_{n}P_{n}(z;r).$

Combining the last two equations with (4.15) and some simplification gives
(4.1).

In order to prove (4.2) we need to consider the following Dodgson condensation
identities:

(4.20) $\begin{split}&\mathscr{D}_{r}\left\\{\begin{matrix}n+2\\\
n+1\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}0&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\}=\mathscr{D}_{r}\left\\{\begin{matrix}0&n+2\\\
n&n+1\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}-\mathscr{D}_{r}\left\\{\begin{matrix}0&n+2\\\
n+1&n+2\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
n&n+1\end{matrix}\right\\},\end{split}$ (4.21)
$\begin{split}&\mathscr{D}_{r}\left\\{\begin{matrix}n+2\\\
n+1\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}0&n+1&n+2\\\
0&n+1&n+2\end{matrix}\right\\}=\mathscr{D}_{r}\left\\{\begin{matrix}0&n+2\\\
0&n+1\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}-\mathscr{D}_{r}\left\\{\begin{matrix}0&n+2\\\
n+1&n+2\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
0&n+1\end{matrix}\right\\}.\end{split}$

Let us write (4.20) as

(4.22) $\frac{\mathscr{D}_{r}\left\\{\begin{matrix}n+2\\\
n+1\end{matrix}\right\\}}{\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}0&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\}=\mathscr{D}_{r}\left\\{\begin{matrix}0&n+2\\\
n&n+1\end{matrix}\right\\}-\frac{\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
n&n+1\end{matrix}\right\\}}{\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}0&n+2\\\
n+1&n+2\end{matrix}\right\\}.$

So from (3.26), (3.28), (3.29), and (2.9) we have

(4.23)
$D^{(r+2)}_{n}Q^{*}_{n+1}(z;r)=D^{(r+2)}_{n}Q^{*}_{n}(z;r+2)-\frac{D^{(r)}_{n}}{D^{(r)}_{n+1}}D^{(r+2)}_{n+1}zQ^{*}_{n}(z;r),$

and thus from (3.3) we get

(4.24)
$Q^{*}_{n+1}(z;r)=Q^{*}_{n}(z;r+2)+\beta^{(r)}_{n}zQ^{*}_{n}(z;r),\qquad\beta^{(r)}_{n}=-\frac{h^{(r+2)}_{n}}{h^{(r)}_{n}}.$

Now, let us write (4.21) as

(4.25) $\frac{\mathscr{D}_{r}\left\\{\begin{matrix}n+2\\\
n+1\end{matrix}\right\\}}{\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}0&n+1&n+2\\\
0&n+1&n+2\end{matrix}\right\\}=\mathscr{D}_{r}\left\\{\begin{matrix}0&n+2\\\
0&n+1\end{matrix}\right\\}-\frac{\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
0&n+1\end{matrix}\right\\}}{\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}0&n+2\\\
n+1&n+2\end{matrix}\right\\}.$

So from (3.26), (3.27), (3.30), and (2.9) we obtain

(4.26)
$D^{(r+1)}_{n}Q^{*}_{n+1}(z;r)=D^{(r+1)}_{n}Q^{*}_{n}(z;r+1)-\frac{D^{(r-1)}_{n}D^{(r+2)}_{n+1}}{D^{(r)}_{n+1}}zQ^{*}_{n}(z;r-1),$

and thus

(4.27)
$Q^{*}_{n+1}(z;r)=Q^{*}_{n}(z;r+1)-\alpha^{(r)}_{n}zQ^{*}_{n}(z;r-1),\qquad\alpha^{(r)}_{n}=\frac{D^{(r-1)}_{n}D^{(r+2)}_{n+1}}{D^{(r)}_{n+1}D^{(r+1)}_{n}}.$

Now, let us find the pure-$n$ recurrence relation for $Q^{*}$s. From (4.24) we
have

(4.28)
$Q^{*}_{n}(z;r+1)=Q^{*}_{n+1}(z;r-1)-\beta^{(r-1)}_{n}zQ^{*}_{n}(z;r-1).$

From (4.27) and (4.28) we get

(4.29)
$Q^{*}_{n+1}(z;r)=Q^{*}_{n+1}(z;r-1)-(\beta^{(r-1)}_{n}+\alpha^{(r)}_{n})zQ^{*}_{n}(z;r-1),$

and immediately

(4.30)
$Q^{*}_{n}(z;r+1)=Q^{*}_{n}(z;r)-(\beta^{(r)}_{n-1}+\alpha^{(r+1)}_{n-1})zQ^{*}_{n-1}(z;r),$

and

(4.31)
$Q^{*}_{n}(z;r+2)=Q^{*}_{n}(z;r+1)-(\beta^{(r+1)}_{n-1}+\alpha^{(r+2)}_{n-1})zQ^{*}_{n-1}(z;r+1).$

Using (4.30) we have

(4.32)
$\begin{split}Q^{*}_{n}(z;r+2)&=Q^{*}_{n}(z;r)-(\beta^{(r)}_{n-1}+\alpha^{(r+1)}_{n-1})zQ^{*}_{n-1}(z;r)-(\beta^{(r+1)}_{n-1}+\alpha^{(r+2)}_{n-1})z\left[Q^{*}_{n-1}(z;r)-(\beta^{(r)}_{n-2}+\alpha^{(r+1)}_{n-2})zQ^{*}_{n-2}(z;r)\right].\end{split}$

Plugging this into (4.24) yields

(4.33)
$\begin{split}Q^{*}_{n+1}(z;r)&=\beta^{(r)}_{n}zQ^{*}_{n}(z;r)+Q^{*}_{n}(z;r)-(\beta^{(r)}_{n-1}+\alpha^{(r+1)}_{n-1})zQ^{*}_{n-1}(z;r)\\\
&-(\beta^{(r+1)}_{n-1}+\alpha^{(r+2)}_{n-1})z\left[Q^{*}_{n-1}(z;r)-(\beta^{(r)}_{n-2}+\alpha^{(r+1)}_{n-2})zQ^{*}_{n-2}(z;r)\right].\end{split}$

Replacing $n$ by $n+2$ and rearranging terms gives the desired recurrence
relation (4.2). ∎

Our second result gives extensions of the second order scalar difference
equations given in Eqs.(2.23,24) of [35] for the $j-2k$ system.

######  Theorem 4.2.

The third order pure-degree recurrence relations for the $j-2k$ polynomials
are given by

(4.34)
$\begin{split}R_{n+3}(z;s)&-\left(z+\varkappa^{(s)}_{n+2}\right)R_{n+2}(z;s)+\left(\varkappa^{(s)}_{n+1}+\rho^{(s-1)}_{n+1}+\rho^{(s)}_{n+2}+\varkappa^{(s)}_{n+2}\right)zR_{n+1}(z;s)\\\
&-\left(\rho^{(s)}_{n+2}+\varkappa^{(s)}_{n+2}\right)\left(\varkappa^{(s)}_{n}+\rho^{(s-1)}_{n}\right)zR_{n}(z;s)=0,\end{split}$

and

(4.35)
$S^{*}_{n+3}(z;s)-\left(\gamma^{(s)}_{n+2}+\gamma^{(s+1)}_{n+1}\right)zS^{*}_{n+2}(z;s)-\left(1-\gamma^{(s+1)}_{n+1}\gamma^{(s)}_{n+1}z^{2}\right)S^{*}_{n+1}(z;s)+\left(\theta^{(s+1)}_{n+1}+\gamma^{(s+1)}_{n+1}\right)zS^{*}_{n}(z;s)=0.$

where

(4.36) $\displaystyle\varkappa^{(s)}_{n}$
$\displaystyle=-\frac{g^{(s-2)}_{n}}{g^{(s)}_{n}},$ (4.37)
$\displaystyle\rho^{(s)}_{n}$
$\displaystyle=\frac{E_{n}^{(s+1)}E_{n+1}^{(s-2)}}{E^{(s)}_{n+1}E^{(s-1)}_{n}},$

(4.38) $\displaystyle\gamma^{(s)}_{n}$
$\displaystyle=-\frac{g^{(s+1)}_{n}}{g^{(s)}_{n}},$ (4.39)
$\displaystyle\theta^{(s)}_{n}$
$\displaystyle=\frac{E^{(s-2)}_{n}E^{(s+1)}_{n+1}}{E^{(s)}_{n+1}E^{(s-1)}_{n}}.$

###### Proof.

In order to prove (4.34) we use the following Dodgson condensation identities

(4.40) $\begin{split}&\mathscr{E}_{s}\left\\{\begin{matrix}n+1\\\
n+2\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}n&n+1&n+2\\\
0&n+1&n+2\end{matrix}\right\\}=\mathscr{E}_{s}\left\\{\begin{matrix}n&n+1\\\
0&n+2\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}-\mathscr{E}_{s}\left\\{\begin{matrix}n&n+1\\\
n+1&n+2\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
0&n+2\end{matrix}\right\\}.\end{split}$ (4.41)
$\begin{split}&\mathscr{E}_{s}\left\\{\begin{matrix}n+1\\\
n+2\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}0&n+1&n+2\\\
0&n+1&n+2\end{matrix}\right\\}=\mathscr{E}_{s}\left\\{\begin{matrix}0&n+1\\\
0&n+2\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}-\mathscr{E}_{s}\left\\{\begin{matrix}0&n+1\\\
n+1&n+2\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
0&n+2\end{matrix}\right\\}.\end{split}$

Write (4.40) as

(4.42) $\frac{\mathscr{E}_{s}\left\\{\begin{matrix}n+1\\\
n+2\end{matrix}\right\\}}{\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}n&n+1&n+2\\\
0&n+1&n+2\end{matrix}\right\\}=\mathscr{E}_{s}\left\\{\begin{matrix}n&n+1\\\
0&n+2\end{matrix}\right\\}-\frac{\mathscr{E}_{s}\left\\{\begin{matrix}n&n+1\\\
n+1&n+2\end{matrix}\right\\}}{\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
0&n+2\end{matrix}\right\\}.$

So using (3.20), (3.21), (3.25), and (2.9) we have

(4.43)
$E^{(s-2)}_{n}R_{n+1}(z;s)=E^{(s-2)}_{n}zR_{n}(z;s-2)-\frac{E_{n}^{(s)}E_{n+1}^{(s-2)}}{E_{n+1}^{(s)}}R_{n}(z;s).$

Therefore from (3.9) we have

(4.44)
$R_{n+1}(z;s)=zR_{n}(z;s-2)+\varkappa^{(s)}_{n}R_{n}(z;s),\qquad\varkappa^{(s)}_{n}=-\frac{g^{(s-2)}_{n}}{g^{(s)}_{n}}.$

Now, lets us write (4.41) as

(4.45) $\frac{\mathscr{E}_{s}\left\\{\begin{matrix}n+1\\\
n+2\end{matrix}\right\\}}{\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}0&n+1&n+2\\\
0&n+1&n+2\end{matrix}\right\\}=\mathscr{E}_{s}\left\\{\begin{matrix}0&n+1\\\
0&n+2\end{matrix}\right\\}-\frac{\mathscr{E}_{s}\left\\{\begin{matrix}0&n+1\\\
n+1&n+2\end{matrix}\right\\}}{\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
0&n+2\end{matrix}\right\\}.$

Therefore from (3.20), (3.22), (3.23), and (2.9) we arrive at

(4.46)
$E^{(s-1)}_{n}R_{n+1}(z;s)=E^{(s-1)}_{n}zR_{n}(z;s-1)-\frac{E_{n}^{(s+1)}E_{n+1}^{(s-2)}}{E^{(s)}_{n+1}}R_{n}(z;s+1),$

and thus

(4.47)
$R_{n+1}(z;s)=zR_{n}(z;s-1)-\rho^{(s)}_{n}R_{n}(z;s+1),\qquad\rho^{(s)}_{n}=\frac{E_{n}^{(s+1)}E_{n+1}^{(s-2)}}{E^{(s)}_{n+1}E^{(s-1)}_{n}}.$

From (4.47) we have

(4.48) $R_{n+2}(z;s)=zR_{n+1}(z;s-1)-\rho^{(s)}_{n+1}R_{n+1}(z;s+1),$

and by writing (4.47) for $s-1$ we obtain

(4.49) $zR_{n}(z;s-2)=R_{n+1}(z;s-1)+\rho^{(s-1)}_{n}R_{n}(z;s).$

Combining this with (4.44) yields

(4.50)
$R_{n+1}(z;s-1)=R_{n+1}(z;s)-(\varkappa^{(s)}_{n}+\rho^{(s-1)}_{n})R_{n}(z;s).$

Plugging this into (4.48) gives

(4.51)
$R_{n+2}(z;s)=z(R_{n+1}(z;s)-(\varkappa^{(s)}_{n}+\rho^{(s-1)}_{n})R_{n}(z;s))-\rho^{(s)}_{n+1}R_{n+1}(z;s+1).$

Now, write (4.44) for $s+1$ to get

(4.52) $R_{n+1}(z;s+1)=zR_{n}(z;s-1)+\varkappa^{(s+1)}_{n}R_{n}(z;s+1).$

After combining the last two equations we arrive at

(4.53)
$R_{n+2}(z;s)=z(R_{n+1}(z;s)-(\varkappa^{(s)}_{n}+\rho^{(s-1)}_{n})R_{n}(z;s))-\rho^{(s)}_{n+1}zR_{n}(z;s-1)-\rho^{(s)}_{n+1}\varkappa^{(s+1)}_{n}R_{n}(z;s+1).$

Note that from (4.47) we have

(4.54)
$R_{n}(z;s+1)=\frac{1}{\rho^{(s)}_{n}}\left[zR_{n}(z;s-1)-R_{n+1}(z;s)\right].$

Combine the last two equations to get

(4.55)
$\begin{split}R_{n+2}(z;s)&=z(R_{n+1}(z;s)-(\varkappa^{(s)}_{n}+\rho^{(s-1)}_{n})R_{n}(z;s))-\rho^{(s)}_{n+1}zR_{n}(z;s-1)-\frac{\rho^{(s)}_{n+1}\varkappa^{(s+1)}_{n}}{\rho^{(s)}_{n}}\left[zR_{n}(z;s-1)-R_{n+1}(z;s)\right].\end{split}$

Rearranging terms and using
$\rho^{(s)}_{n+1}\varkappa^{(s+1)}_{n}=\rho^{(s)}_{n}\varkappa^{(s)}_{n+1}$
yields

(4.56)
$\begin{split}R_{n+2}(z;s)&=\left(z+\varkappa^{(s)}_{n+1}\right)R_{n+1}(z;s)-(\varkappa^{(s)}_{n}+\rho^{(s-1)}_{n})zR_{n}(z;s)-\left(\rho^{(s)}_{n+1}+\varkappa^{(s)}_{n+1}\right)zR_{n}(z;s-1).\end{split}$

By writing (4.50) for $n-1$ we obtain

(4.57)
$R_{n}(z;s-1)=R_{n}(z;s)-(\varkappa^{(s)}_{n-1}+\rho^{(s-1)}_{n-1})R_{n-1}(z;s).$

Now, we combine the last two equations to arrive at

(4.58)
$\begin{split}R_{n+2}(z;s)&=\left(z+\varkappa^{(s)}_{n+1}\right)R_{n+1}(z;s)-(\varkappa^{(s)}_{n}+\rho^{(s-1)}_{n})zR_{n}(z;s)-\left(\rho^{(s)}_{n+1}+\varkappa^{(s)}_{n+1}\right)\left(zR_{n}(z;s)-(\varkappa^{(s)}_{n-1}+\rho^{(s-1)}_{n-1})zR_{n-1}(z;s)\right).\end{split}$

regroup terms and replace $n$ by $n+1$ to arrive at the desired recurrence
relation (4.34).

Finally, proving (4.35) demands using the following Dodgson condensation
identities

(4.59) $\begin{split}&\mathscr{E}_{s}\left\\{\begin{matrix}n+2\\\
n+1\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}0&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\}=\mathscr{E}_{s}\left\\{\begin{matrix}0&n+2\\\
n&n+1\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}-\mathscr{E}_{s}\left\\{\begin{matrix}0&n+2\\\
n+1&n+2\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
n&n+1\end{matrix}\right\\}.\end{split}$

(4.60) $\begin{split}&\mathscr{E}_{s}\left\\{\begin{matrix}n+2\\\
n+1\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}0&n+1&n+2\\\
0&n+1&n+2\end{matrix}\right\\}=\mathscr{E}_{s}\left\\{\begin{matrix}0&n+2\\\
0&n+1\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}-\mathscr{E}_{s}\left\\{\begin{matrix}0&n+2\\\
n+1&n+2\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
0&n+1\end{matrix}\right\\}.\end{split}$

Write (4.59) as

(4.61) $\frac{\mathscr{E}_{s}\left\\{\begin{matrix}n+2\\\
n+1\end{matrix}\right\\}}{\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}0&n+1&n+2\\\
n&n+1&n+2\end{matrix}\right\\}=\mathscr{E}_{s}\left\\{\begin{matrix}0&n+2\\\
n&n+1\end{matrix}\right\\}-\frac{\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
n&n+1\end{matrix}\right\\}}{\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}0&n+2\\\
n+1&n+2\end{matrix}\right\\}.$

So we have

(4.62)
$E^{(s+1)}_{n}S^{*}_{n+1}(z;s)=E^{(s+1)}_{n}S^{*}_{n}(z;s+1)-\frac{E^{(s)}_{n}}{E^{(s)}_{n+1}}E^{(s+1)}_{n+1}zS^{*}_{n}(z;s).$

and thus from (3.9) we have

(4.63)
$S^{*}_{n+1}(z;s)=S^{*}_{n}(z;s+1)+\gamma^{(s)}_{n}zS^{*}_{n}(z;s),\qquad\gamma^{(s)}_{n}=-\frac{g^{(s+1)}_{n}}{g^{(s)}_{n}}.$

Now, rewrite (4.60) as

(4.64) $\frac{\mathscr{E}_{s}\left\\{\begin{matrix}n+2\\\
n+1\end{matrix}\right\\}}{\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}0&n+1&n+2\\\
0&n+1&n+2\end{matrix}\right\\}=\mathscr{E}_{s}\left\\{\begin{matrix}0&n+2\\\
0&n+1\end{matrix}\right\\}-\frac{\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
0&n+1\end{matrix}\right\\}}{\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
n+1&n+2\end{matrix}\right\\}}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}0&n+2\\\
n+1&n+2\end{matrix}\right\\}.$

So

(4.65)
$E^{(s-1)}_{n}S^{*}_{n+1}(z;s)=E^{(s-1)}_{n}S^{*}_{n}(z;s-1)-\frac{E^{(s-2)}_{n}E^{(s+1)}_{n+1}}{E^{(s)}_{n+1}}zS^{*}_{n}(z;s-2),$

and thus

(4.66)
$S^{*}_{n+1}(z;s)=S^{*}_{n}(z;s-1)-\theta^{(s)}_{n}zS^{*}_{n}(z;s-2),\qquad\theta^{(s)}_{n}=\frac{E^{(s-2)}_{n}E^{(s+1)}_{n+1}}{E^{(s)}_{n+1}E^{(s-1)}_{n}}.$

Now let us find the pure degree-recurrence relations for $S^{*}$’s. Writing
(4.63) for $n+1$ we get

(4.67)
$S^{*}_{n+2}(z;s)=S^{*}_{n+1}(z;s+1)+\gamma^{(s)}_{n+1}zS^{*}_{n+1}(z;s),$

Now we write (4.66) for $s+1$:

(4.68)
$S^{*}_{n+1}(z;s+1)=S^{*}_{n}(z;s)-\theta^{(s+1)}_{n}zS^{*}_{n}(z;s-1).$

Combining the last two equations yields

(4.69)
$S^{*}_{n+2}(z;s)=S^{*}_{n}(z;s)-\theta^{(s+1)}_{n}zS^{*}_{n}(z;s-1)+\gamma^{(s)}_{n+1}zS^{*}_{n+1}(z;s).$

Write this for $n\mapsto n+1$:

(4.70)
$S^{*}_{n+3}(z;s)=S^{*}_{n+1}(z;s)-\theta^{(s+1)}_{n+1}zS^{*}_{n+1}(z;s-1)+\gamma^{(s)}_{n+2}zS^{*}_{n+2}(z;s),$

and write (4.63) for $r\mapsto r-1$ to obtain

(4.71)
$S^{*}_{n+1}(z;s-1)=S^{*}_{n}(z;s)+\gamma^{(s-1)}_{n}zS^{*}_{n}(z;s-1).$

Combining the last two equations gives

(4.72)
$S^{*}_{n+3}(z;s)=S^{*}_{n+1}(z;s)-\theta^{(s+1)}_{n+1}zS^{*}_{n}(z;s)-\theta^{(s+1)}_{n+1}\gamma^{(s-1)}_{n}z^{2}S^{*}_{n}(z;s-1)+\gamma^{(s)}_{n+2}zS^{*}_{n+2}(z;s).$

Now write (4.66) for $s\mapsto s+1$ and solve for $zS_{n}(z;s-1)$:

(4.73)
$zS^{*}_{n}(z;s-1)=\frac{1}{\theta^{(s+1)}_{n}}\left(S^{*}_{n}(z;s)-S^{*}_{n+1}(z;s+1)\right).$

Combine the last two equations to get

(4.74)
$S^{*}_{n+3}(z;s)=S^{*}_{n+1}(z;s)-\theta^{(s+1)}_{n+1}zS^{*}_{n}(z;s)-\frac{\theta^{(s+1)}_{n+1}\gamma^{(s-1)}_{n}z}{\theta^{(s+1)}_{n}}\left(S^{*}_{n}(z;s)-S^{*}_{n+1}(z;s+1)\right)+\gamma^{(s)}_{n+2}zS^{*}_{n+2}(z;s).$

By using
$\theta^{(s+1)}_{n+1}\gamma^{(s-1)}_{n}=\theta^{(s+1)}_{n}\gamma^{(s+1)}_{n+1}$
and rearranging terms we obtain

(4.75)
$S^{*}_{n+3}(z;s)=\gamma^{(s)}_{n+2}zS^{*}_{n+2}(z;s)+S^{*}_{n+1}(z;s)-\left(\theta^{(s+1)}_{n+1}+\gamma^{(s+1)}_{n+1}\right)zS^{*}_{n}(z;s)+\gamma^{(s+1)}_{n+1}zS^{*}_{n+1}(z;s+1).$

Write (4.63) for $n\mapsto n+1$ and solve for $S^{*}_{n+1}(z;s+1)$:

(4.76)
$S^{*}_{n+1}(z;s+1)=S^{*}_{n+2}(z;s)-\gamma^{(s)}_{n+1}zS^{*}_{n+1}(z;s).$

Now, we combine the last two equations and regroup the terms to obtain (4.35).
∎

######  Remark 4.3 (Mixed Recurrence Relations).

Recall that we can replace a $2j-k$ polynomial in terms of a $j-2k$ polynomial
using Theorem 3.3. Using such replacements in several identities found in this
section, we can arrive at recurrence relations involving both $2j-k$ and
$j-2k$ polynomials. To that end, (4.11) can be written as

(4.77)
$P_{n+1}(z;r)=\frac{(-1)^{n}E_{n}^{(r+n-3)}}{D_{n}^{(r-1)}}zS^{*}_{n}(z;r+n-3)+\delta^{(r)}_{n}P_{n}(z;r).$

Also (4.14) can be written as one of the following three identities

(4.78) $\displaystyle P_{n+1}(z;r)$
$\displaystyle=\frac{(-1)^{n}E_{n}^{(r+n-1)}}{D_{n}^{(r+1)}}zS^{*}_{n}(z;r+n-1)-\eta^{(r)}_{n}P_{n}(z;r+2),$
(4.79) $\displaystyle P_{n+1}(z;r)$
$\displaystyle=zP_{n}(z;r+1)-\eta^{(r)}_{n}\frac{(-1)^{n}E_{n}^{(r+n)}}{D_{n}^{(r+2)}}S^{*}_{n}(z;r+n),$
(4.80) $\displaystyle P_{n+1}(z;r)$
$\displaystyle=\frac{(-1)^{n}E_{n}^{(r+n-1)}}{D_{n}^{(r+1)}}zS^{*}_{n}(z;r+n-1)-\eta^{(r)}_{n}\frac{(-1)^{n}E_{n}^{(r+n)}}{D_{n}^{(r+2)}}S^{*}_{n}(z;r+n).$

In an identical way we can obtain the following mixed recurrence relations
listed below

(4.81)
$Q^{*}_{n+1}(z;r)=\frac{(-1)^{n}E_{n}^{(r+n+3)}}{D_{n}^{(r+2)}}R_{n}(z;r+n+3)+\beta^{(r)}_{n}zQ^{*}_{n}(z;r),$
(4.82)
$Q^{*}_{n+1}(z;r)=\frac{(-1)^{n}E_{n}^{(r+n+2)}}{D_{n}^{(r+1)}}R_{n}(z;r+n+2)-\alpha^{(r)}_{n}zQ^{*}_{n}(z;r-1),$
(4.83)
$Q^{*}_{n+1}(z;r)=Q^{*}_{n}(z;r+1)-\alpha^{(r)}_{n}\frac{(-1)^{n}E_{n}^{(r+n)}}{D_{n}^{(r-1)}}zR_{n}(z;r+n),$
(4.84)
$Q^{*}_{n+1}(z;r)=\frac{(-1)^{n}E_{n}^{(r+n+2)}}{D_{n}^{(r+1)}}R_{n}(z;r+n+2)-\alpha^{(r)}_{n}\frac{(-1)^{n}E_{n}^{(r+n)}}{D_{n}^{(r-1)}}zR_{n}(z;r+n),$
(4.85)
$R_{n+1}(z;s)=\frac{(-1)^{n}D_{n}^{(s-n-3)}}{E_{n}^{(s-2)}}zQ^{*}_{n}(z;s-n-3)+\varkappa^{(s)}_{n}R_{n}(z;s),$
(4.86)
$R_{n+1}(z;s)=\frac{(-1)^{n}D_{n}^{(s-n-2)}}{E_{n}^{(s-1)}}zQ^{*}_{n}(z;s-n-2)-\rho^{(s)}_{n}R_{n}(z;s+1),$
(4.87)
$R_{n+1}(z;s)=zR_{n}(z;s-1)-\rho^{(s)}_{n}\frac{(-1)^{n}D_{n}^{(s-n)}}{E_{n}^{(s+1)}}Q^{*}_{n}(z;s-n),$
(4.88)
$R_{n+1}(z;s)=\frac{(-1)^{n}D_{n}^{(s-n-2)}}{E_{n}^{(s-1)}}zQ^{*}_{n}(z;s-n-2)-\rho^{(s)}_{n}\frac{(-1)^{n}D_{n}^{(s-n)}}{E_{n}^{(s+1)}}Q^{*}_{n}(z;s-n),$

(4.89)
$S^{*}_{n+1}(z;s)=\frac{(-1)^{n}D^{(s-n+3)}_{n}}{E_{n}^{(s+1)}}P_{n}(z;s-n+3)+\gamma^{(s)}_{n}zS^{*}_{n}(z;s),$

(4.90)
$S^{*}_{n+1}(z;s)=\frac{(-1)^{n}D^{(s-n+1)}_{n}}{E_{n}^{(s-1)}}P_{n}(z;s-n+1)-\theta^{(s)}_{n}zS^{*}_{n}(z;s-2),$

(4.91)
$S^{*}_{n+1}(z;s)=S^{*}_{n}(z;s-1)-\theta^{(s)}_{n}\frac{(-1)^{n}D^{(s-n)}_{n}}{E_{n}^{(s-2)}}zP_{n}(z;s-n),$

and

(4.92)
$S^{*}_{n+1}(z;s)=\frac{(-1)^{n}D^{(s-n+1)}_{n}}{E_{n}^{(s-1)}}P_{n}(z;s-n+1)-\theta^{(s)}_{n}\frac{(-1)^{n}D^{(s-n)}_{n}}{E_{n}^{(s-2)}}zP_{n}(z;s-n).$

Here the identities (4.81), (4.85), and (4.89) are respectively equivalent to
(4.24), (4.44), and (4.63) . Identities (4.82), (4.83) and (4.84) are
equivalent to (4.27), identities (4.86), (4.87) and (4.88) are equivalent to
(4.47), and identities (4.90), (4.91) and (4.92) are equivalent to (4.66).

######  Theorem 4.4.

The third order pure-offset recurrence relations for the $2j-k$ and $j-2k$
polynomials are given by

(4.93)
$\eta^{(r+1)}_{n}P_{n}(z;r+3)-zP_{n}(z;r+2)+\delta^{(r+1)}_{n}P_{n}(z;r+1)+zP_{n}(z;r)=0,$
(4.94)
$Q^{*}_{n}(z;r+3)-Q^{*}_{n}(z;r+2)+\beta^{(r+1)}_{n}zQ^{*}_{n}(z;r+1)+\alpha^{(r+1)}_{n}zQ^{*}_{n}(z;r)=0,$
(4.95)
$\rho^{(s+2)}_{n}R_{n}(z;s+3)+\varkappa^{(s+2)}_{n}R_{n}(z;s+2)-zR_{n}(z;s+1)+zR_{n}(z;s)=0,$

and

(4.96)
$S^{*}_{n}(z;s+3)+\gamma^{(s+2)}_{n}zS^{*}_{n}(z;s+2)-S^{*}_{n}(z;s+1)+\theta^{(s+2)}_{n}zS^{*}_{n}(z;s)=0.$

###### Proof.

Combining (4.11) and (4.14) we will get

(4.97)
$\eta^{(r)}_{n}P_{n}(z;r+2)-zP_{n}(z;r+1)+zP_{n}(z;r-1)+\delta^{(r)}_{n}P_{n}(z;r)=0.$

Also from (4.24) and (4.27) we obtain

(4.98)
$Q^{*}_{n}(z;r+2)-Q^{*}_{n}(z;r+1)+\beta^{(r)}_{n}zQ^{*}_{n}(z;r)+\alpha^{(r)}_{n}zQ^{*}_{n}(z;r-1)=0.$

writing (4.97) and (4.98) under the replacement $r\mapsto r+1$ respectively
gives (4.93) and (4.94). From (4.44) and (4.47) we have

(4.99)
$zR_{n}(z;s-2)+\varkappa^{(s)}_{n}R_{n}(z;s)-zR_{n}(z;s-1)+\rho^{(s)}_{n}R_{n}(z;s+1)=0.$

From (4.63) and (4.66) we find

(4.100)
$S^{*}_{n}(z;s+1)+\gamma^{(s)}_{n}zS^{*}_{n}(z;s)-S^{*}_{n}(z;s-1)+\theta^{(s)}_{n}zS^{*}_{n}(z;s-2)=0.$

The replacement $s\mapsto s+2$ in (4.99) and (4.100) respectively gives (4.95)
and (4.96). ∎

###  4.1. Equivalent Dodgson Condensation identities

If one starts with

(4.101) $\begin{split}&\mathscr{D}_{r}\left\\{\begin{matrix}0\\\
n+2\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}0&n+1&n+2\\\
0&n+1&n+2\end{matrix}\right\\}=\mathscr{D}_{r}\left\\{\begin{matrix}0&n+1\\\
0&n+2\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}0&n+2\\\
n+1&n+2\end{matrix}\right\\}-\mathscr{D}_{r}\left\\{\begin{matrix}0&n+1\\\
n+1&n+2\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}0&n+2\\\
0&n+2\end{matrix}\right\\},\end{split}$

after similar simplifications one arrives at

(4.102)
$P_{n+1}(z;r+2)=zP_{n}(z;r+1)-\frac{h^{(r+1)}_{n}}{h^{(r+2)}_{n}}P_{n}(z;r+2).$

But this is equivalent to (4.11) as one gets the above equation from (4.11)
under the replacement $r\mapsto r+2$. Similarly, If we start with

(4.103) $\begin{split}&\mathscr{D}_{r}\left\\{\begin{matrix}n+2\\\
0\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}0&n+1&n+2\\\
0&n+1&n+2\end{matrix}\right\\}=\mathscr{D}_{r}\left\\{\begin{matrix}0&n+2\\\
0&n+1\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
0&n+2\end{matrix}\right\\}-\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
0&n+1\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}0&n+2\\\
0&n+2\end{matrix}\right\\}.\end{split}$

After simplifications, this translates to

(4.104)
$Q^{*}_{n+1}(z;r-1)=Q^{*}_{n}(z;r+1)-\frac{h^{(r+1)}_{n}}{h^{(r-1)}_{n}}zQ^{*}_{n}(z;r-1).$

However, this is equivalent to (4.24) if we replace $r$ by $r-1$ in (4.24).
Also note that

(4.105) $\begin{split}&\mathscr{E}_{s}\left\\{\begin{matrix}0\\\
n+2\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}0&n+1&n+2\\\
0&n+1&n+2\end{matrix}\right\\}=\mathscr{E}_{s}\left\\{\begin{matrix}0&n+1\\\
0&n+2\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}0&n+2\\\
n+1&n+2\end{matrix}\right\\}-\mathscr{E}_{s}\left\\{\begin{matrix}0&n+1\\\
n+1&n+2\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}0&n+2\\\
0&n+2\end{matrix}\right\\}.\end{split}$

This reduces after simplifications to

(4.106)
$R_{n+1}(z;s+1)=zR_{n}(z;s-1)-\frac{g^{(s-1)}_{n}}{g^{(s+1)}_{n}}R_{n}(z;s+1),$

which is equivalent to (4.44) if one replaces $s$ with $s+1$ in (4.44). And
finally

(4.107) $\begin{split}&\mathscr{E}_{s}\left\\{\begin{matrix}n+2\\\
0\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}0&n+1&n+2\\\
0&n+1&n+2\end{matrix}\right\\}=\mathscr{E}_{s}\left\\{\begin{matrix}0&n+2\\\
0&n+1\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
0&n+2\end{matrix}\right\\}-\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
0&n+1\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}0&n+2\\\
0&n+2\end{matrix}\right\\}\end{split}$

reduces after simplifications to

(4.108)
$S^{*}_{n+1}(z;s-2)=S^{*}_{n}(z;s-1)-\frac{g^{(s-1)}_{n}}{g^{(s-2)}_{n}}zS^{*}_{n}(z;s-2),$

which is equivalent to (4.63) if one replaces $s$ with $s-2$ in (4.63). In a
similar fashion one can find four other Dodgson Condensation identities which
give rise to recurrence relations equivalent to (4.14), (4.27), (4.47), and
(4.66), these Dodgson Condensation identities are respectively given by

(4.109) $\displaystyle\mathscr{D}_{r}\left\\{\begin{matrix}0\\\
n+2\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}0&1&n+2\\\
0&n+1&n+2\end{matrix}\right\\}$
$\displaystyle=\mathscr{D}_{r}\left\\{\begin{matrix}0&1\\\
0&n+2\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}0&n+2\\\
n+1&n+2\end{matrix}\right\\}$
$\displaystyle-\mathscr{D}_{r}\left\\{\begin{matrix}0&1\\\
n+1&n+2\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}0&n+2\\\
0&n+2\end{matrix}\right\\},$ (4.110)
$\displaystyle\mathscr{D}_{r}\left\\{\begin{matrix}n+2\\\
0\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}0&n+1&n+2\\\
0&1&n+2\end{matrix}\right\\}$
$\displaystyle=\mathscr{D}_{r}\left\\{\begin{matrix}0&n+2\\\
0&1\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
0&n+2\end{matrix}\right\\}$
$\displaystyle-\mathscr{D}_{r}\left\\{\begin{matrix}n+1&n+2\\\
0&1\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}0&n+2\\\
0&n+2\end{matrix}\right\\},$ (4.111)
$\displaystyle\mathscr{E}_{s}\left\\{\begin{matrix}0\\\
n+2\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}0&1&n+2\\\
0&n+1&n+2\end{matrix}\right\\}$
$\displaystyle=\mathscr{E}_{s}\left\\{\begin{matrix}0&1\\\
0&n+2\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}0&n+2\\\
n+1&n+2\end{matrix}\right\\}$
$\displaystyle-\mathscr{E}_{s}\left\\{\begin{matrix}0&1\\\
n+1&n+2\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}0&n+2\\\
0&n+2\end{matrix}\right\\},$ (4.112)
$\displaystyle\mathscr{E}_{s}\left\\{\begin{matrix}n+2\\\
0\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}0&n+1&n+2\\\
0&1&n+2\end{matrix}\right\\}$
$\displaystyle=\mathscr{E}_{s}\left\\{\begin{matrix}0&n+2\\\
0&1\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
0&n+2\end{matrix}\right\\}$
$\displaystyle-\mathscr{E}_{s}\left\\{\begin{matrix}n+1&n+2\\\
0&1\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}0&n+2\\\
0&n+2\end{matrix}\right\\}.$

###  4.2. Relationships of polynomial tails, recurrence coefficients and
determinants

There are many redundancies amongst the various coefficients that have been
defined thus far. In this subsection we provide the necessary relationships
between these coefficients to remove the redundancy, but also to link these to
certain polynomial coefficients, in particular the tail coefficients, and to
the determinants and thus the norms of the polynomials. Such relations form an
algorithmic path to compute these norms.

######  Lemma 4.5.

There exist interrelationships between the recurrence coefficients
$\alpha^{(r)}_{n}$, $\beta^{(r)}_{n}$, $\delta^{(r)}_{n}$, and
$\eta^{(r)}_{n}$ and also between $\varkappa^{(s)}_{n}$, $\rho^{(s)}_{n}$,
$\gamma^{(s)}_{n}$, and $\theta^{(s)}_{n}$ as described by

(4.113) $\displaystyle\delta^{(r)}_{n}\alpha^{(r)}_{n}$
$\displaystyle=\beta^{(r)}_{n}\eta^{(r)}_{n},$

(4.114) $\displaystyle\varkappa^{(s)}_{n}\theta^{(s)}_{n}$
$\displaystyle=\gamma^{(s)}_{n}\rho^{(s)}_{n}.$

###### Proof.

These relationships simply follow from the definitions of the recurrence
relations in terms of the $2j-k$ and $j-2k$ determinants. ∎

######  Lemma 4.6.

The $2j-k$ and the $j-2k$ systems each have only two independent recurrence
coefficients due to the following interrelationships:

(4.115) $\displaystyle\beta^{(r)}_{n}$
$\displaystyle=-\frac{1}{\delta^{(r+1)}_{n}\delta^{(r+2)}_{n}},$ (4.116)
$\displaystyle\alpha^{(r)}_{n}$
$\displaystyle=-\frac{\eta^{(r)}_{n}}{\delta^{(r)}_{n}\delta^{(r+1)}_{n}\delta^{(r+2)}_{n}},$

(4.117) $\displaystyle\varkappa^{(s)}_{n}$
$\displaystyle=-\frac{1}{\gamma^{(s-1)}_{n}\gamma^{(s-2)}_{n}},$ (4.118)
$\displaystyle\rho^{(s)}_{n}$
$\displaystyle=-\frac{\theta^{(s)}_{n}}{\gamma^{(s)}_{n}\gamma^{(s-1)}_{n}\gamma^{(s-2)}_{n}}.$

###### Proof.

From (4.24) we have

(4.119)
$\beta^{(r)}_{n}=-\frac{h^{(r+2)}_{n}}{h^{(r)}_{n}}=-\frac{h^{(r+1)}_{n}}{h^{(r)}_{n}}\frac{h^{(r+2)}_{n}}{h^{(r+1)}_{n}}=-\frac{1}{\delta^{(r+1)}_{n}\delta^{(r+2)}_{n}},$

by the definition of $\delta^{(r)}_{n}$ in (4.11). From (4.27) we have

(4.120)
$\alpha^{(r)}_{n}=\frac{D^{(r-1)}_{n}D^{(r+2)}_{n+1}}{D^{(r)}_{n+1}D^{(r+1)}_{n}}=\frac{D^{(r-1)}_{n}D^{(r+2)}_{n}}{D^{(r)}_{n}D^{(r+1)}_{n}}\frac{h^{(r+2)}_{n}}{h^{(r)}_{n}}.$

Also from (4.14) we have

(4.121)
$\eta^{(r)}_{n}=\frac{D_{n}^{(r+2)}D_{n+1}^{(r-1)}}{D^{(r)}_{n+1}D^{(r+1)}_{n}}=\frac{D_{n}^{(r+2)}D_{n}^{(r-1)}}{D^{(r)}_{n}D^{(r+1)}_{n}}\frac{h^{(r-1)}_{n}}{h^{(r)}_{n}}.$

Combining the last two equations gives

(4.122)
$\alpha^{(r)}_{n}=\frac{h^{(r+2)}_{n}}{h^{(r-1)}_{n}}\eta^{(r)}_{n}=\frac{h^{(r+2)}_{n}}{h^{(r+1)}_{n}}\frac{h^{(r+1)}_{n}}{h^{(r)}_{n}}\frac{h^{(r)}_{n}}{h^{(r-1)}_{n}}\eta^{(r)}_{n}=-\frac{\eta^{(r)}_{n}}{\delta^{(r)}_{n}\delta^{(r+1)}_{n}\delta^{(r+2)}_{n}},$

again by the definition of $\delta^{(r)}_{n}$ in (4.11). The relations (4.117)
and (4.118) can be proven similarly. ∎

The following result relates the tail coefficients of the polynomials to
products of certain recurrence relation coefficients.

######  Lemma 4.7.

The polynomial tails can be expressed in terms of the recurrence coefficients
as follows

(4.123) $\displaystyle P_{n}(0;r)$
$\displaystyle=\prod^{n-1}_{\ell=0}\delta^{(r)}_{\ell},$ (4.124)
$\displaystyle Q_{n}(0;r)$
$\displaystyle=\prod^{n-1}_{\ell=0}\beta^{(r)}_{\ell},$

(4.125) $\displaystyle R_{n}(0;s)$
$\displaystyle=\prod^{n-1}_{\ell=0}\varkappa^{(s)}_{\ell},$ (4.126)
$\displaystyle S_{n}(0;s)$
$\displaystyle=\prod^{n-1}_{\ell=0}\gamma^{(s)}_{\ell}.$

###### Proof.

These identities are immediate consequences of the recurrence relations
(4.11), (4.24), (4.44), and (4.63). For example evaluating (4.11) at $z=0$
gives $P_{n+1}(0;r)=\delta^{(r)}_{n}P_{n}(0;r)$ and thus (4.123) follows
because $P_{0}(0;r)=1$. Now, if we match the coefficients of $z^{n+1}$ in
(4.24) we obtain $Q_{n+1}(0;r)=\beta^{(r)}_{n}Q_{n}(0;r)$, which yields
(4.124) because $Q_{0}(0;r)=1$. The identities (4.125) and (4.126) can be
shown similarly using respectively (4.44), and (4.63). ∎

The next set of relations are just the inverse relations to the ones in the
preceding Lemma.

######  Corollary 4.7.1.

The polynomial tails satisfy the following pure-$n$ recurrence relations

(4.127) $\displaystyle P_{n+1}(0;r)$
$\displaystyle=\delta^{(r)}_{n}P_{n}(0;r),$ (4.128) $\displaystyle
Q_{n+1}(0;r)$ $\displaystyle=\beta^{(r)}_{n}Q_{n}(0;r),$

(4.129) $\displaystyle R_{n+1}(0;s)$
$\displaystyle=\varkappa^{(s)}_{n}R_{n}(0;s),$ (4.130) $\displaystyle
S_{n+1}(0;s)$ $\displaystyle=\gamma^{(r)}_{n}S_{n}(0;s).$

######  Corollary 4.7.2.

The following relationships hold between the $2j-k$ and $j-2k$ polynomial
tails

(4.131) $Q_{n}(0;r)=\frac{(-1)^{n}}{P_{n}(0;r+1)P_{n}(0;r+2)},$

(4.132) $R_{n}(0;s)=\frac{(-1)^{n}}{S_{n}(0;s-1)S_{n}(0;s-2)}.$

###### Proof.

From (4.115) and (4.117) we have

(4.133)
$\prod_{\ell=0}^{n-1}\beta^{(r)}_{\ell}=\frac{(-1)^{n}}{\prod_{\ell=0}^{n-1}\delta^{(r+1)}_{\ell}\prod_{\ell=0}^{n-1}\delta^{(r+2)}_{\ell}},\quad\text{and}\quad\prod_{\ell=0}^{n-1}\varkappa^{(s)}_{\ell}=\frac{(-1)^{n}}{\prod_{\ell=0}^{n-1}\gamma^{(s-1)}_{\ell}\prod_{\ell=0}^{n-1}\gamma^{(s-2)}_{\ell}}.$

Now (4.131) and (4.132) follow from (4.123), (4.124), (4.125), and (4.126). ∎

The following result relates the tail coefficients of the polynomials to
certain ratios of determinants. This is the extension of the Toeplitz
relations for the reflection or Verblunsky coefficients given by Eqs. (2.19)
of [35].

######  Lemma 4.8.

The polynomial tails can be expressed in terms of the ratios of $2j-k$ and
$j-2k$ determinants as follows

(4.134) $\displaystyle P_{n}(0;r)$
$\displaystyle=(-1)^{n}\frac{D^{(r-1)}_{n}}{D^{(r)}_{n}},$ (4.135)
$\displaystyle Q_{n}(0;r)$
$\displaystyle=(-1)^{n}\frac{D^{(r+2)}_{n}}{D^{(r)}_{n}},$

(4.136) $\displaystyle R_{n}(0;s)$
$\displaystyle=(-1)^{n}\frac{E^{(s-2)}_{n}}{E^{(s)}_{n}},$ (4.137)
$\displaystyle S_{n}(0;s)$
$\displaystyle=(-1)^{n}\frac{E^{(s+1)}_{n}}{E^{(s)}_{n}}.$

###### Proof.

These are immediate consequences of the identities (3.1), (3.2), (3.7), and
(3.8). ∎

######  Lemma 4.9.

The polynomial tails satisfy the following pure offset recurrence relations

(4.138) $\displaystyle P_{n}(0;r+2)$
$\displaystyle=-\frac{\delta^{(r)}_{n}}{\eta^{(r)}_{n}}P_{n}(0;r),$ (4.139)
$\displaystyle Q_{n}(0;r-1)$
$\displaystyle=-\frac{\beta^{(r)}_{n}}{\alpha^{(r)}_{n}}Q_{n}(0;r),$

(4.140) $\displaystyle R_{n}(0;s+1)$
$\displaystyle=-\frac{\varkappa^{(s)}_{n}}{\rho^{(s)}_{n}}R_{n}(0;s),$ (4.141)
$\displaystyle S_{n}(0;s-2)$
$\displaystyle=-\frac{\gamma^{(s)}_{n}}{\theta^{(s)}_{n}}S_{n}(0;s).$

###### Proof.

These are immediate consequences of the identities (4.93), (4.94), (4.95), and
(4.96) by matching the constant terms and coefficients of $z^{n+1}$. ∎

Lemma 4.7 establishes the construction of the tails of the orthogonal
polynomials, provided the data of $2j-k$ and $j-2k$ determinants (and thus the
constants $\delta_{n}^{(r)}$,$\beta_{n}^{(r)}$,$\varkappa_{n}^{(r)}$, and
$\gamma_{n}^{(r)}$) are given. The following theorem establishes this in the
reverse direction: how to construct the $2j-k$ and $j-2k$ determinants from
the data given as the tails of the corresponding orthogonal polynomials. This
result is the extension of the Toeplitz result given by Eq.(2.20) of [35].

######  Theorem 4.10.

The $2j-k$ and $j-2k$ determinants can be constructed from the tails of the
orthogonal polynomials as

(4.142)
$D^{(r)}_{n}=w^{n}_{r}\prod_{\ell=0}^{n-1}\frac{1}{P_{\ell}(0;r+1)}\prod_{\nu=1}^{\ell}\left(\frac{P_{\nu}(0;r)}{P_{\nu}(0;r+2)}-1\right),$

and

(4.143)
$E^{(s)}_{n}=w^{n}_{s}\prod_{\ell=0}^{n-1}\frac{1}{S_{\ell}(0;s-1)}\prod_{\nu=1}^{\ell}\left(\frac{S_{\nu}(0;s)}{S_{\nu}(0;s-2)}-1\right).$

###### Proof.

Let $1\leq\nu\leq n-1$ be a fixed integer and let $\bm{\mathscr{D}}_{r}$ be
the $(\nu+3)\times(\nu+3)$ matrix given by (2.3). Now consider the following
Dodgson condensation identity:

(4.144) $\begin{split}&\mathscr{D}_{r}\left\\{\begin{matrix}\nu+2\\\
\nu+2\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}0&\nu+1&\nu+2\\\
0&\nu+1&\nu+2\end{matrix}\right\\}=\mathscr{D}_{r}\left\\{\begin{matrix}0&\nu+2\\\
0&\nu+2\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}\nu+1&\nu+2\\\
\nu+1&\nu+2\end{matrix}\right\\}-\mathscr{D}_{r}\left\\{\begin{matrix}0&\nu+2\\\
\nu+1&\nu+2\end{matrix}\right\\}\cdot\mathscr{D}_{r}\left\\{\begin{matrix}\nu+1&\nu+2\\\
0&\nu+2\end{matrix}\right\\}.\end{split}$

In view of (2.3), (3.3), (4.134) and (4.135), the above identity can be
written as

(4.145)
$h^{(r)}_{\nu+1}\frac{D^{(r+1)}_{\nu}}{D^{(r)}_{\nu+1}}=\frac{D^{(r+1)}_{\nu+1}}{D^{(r)}_{\nu+1}}-Q_{\nu+1}(0;r)P_{\nu+1}(0;r).$

Using (3.3) again and replacing $\nu\mapsto\nu-1$ yields

(4.146)
$\left(\frac{h^{(r)}_{\nu}}{h^{(r+1)}_{\nu-1}}-1\right)\frac{D^{(r+1)}_{\nu}}{D^{(r)}_{\nu}}=-Q_{\nu}(0;r)P_{\nu}(0;r).$

Now we use (4.134) to rewrite this as

(4.147)
$\frac{h^{(r)}_{\nu}}{h^{(r+1)}_{\nu-1}}=1-(-1)^{\nu}Q_{\nu}(0;r)P_{\nu}(0;r)P_{\nu}(0;r+1)=1-\frac{P_{\nu}(0;r)}{P_{\nu}(0;r+2)},$

where we have used (4.131). Therefore

(4.148)
$\frac{\prod_{\nu=1}^{\ell}h^{(r)}_{\nu}}{\prod_{\nu=1}^{\ell}h^{(r+1)}_{\nu-1}}=\prod_{\nu=1}^{\ell}\left(1-\frac{P_{\nu}(0;r)}{P_{\nu}(0;r+2)}\right).$

From (3.4) we will get

(4.149)
$\frac{h^{(r)}_{\ell}}{h^{(r)}_{0}}\frac{D^{(r)}_{\ell}}{D^{(r+1)}_{\ell}}=\prod_{\nu=1}^{\ell}\left(1-\frac{P_{\nu}(0;r)}{P_{\nu}(0;r+2)}\right).$

Using (4.134) and noticing that $h^{(r)}_{0}=D^{(r)}_{1}=w_{r}$, we have

(4.150)
$h^{(r)}_{\ell}=w_{r}\frac{(-1)^{\ell}}{P_{\ell}(0;r+1)}\prod_{\nu=1}^{\ell}\left(1-\frac{P_{\nu}(0;r)}{P_{\nu}(0;r+2)}\right).$

Finally, using (3.4) we obtain (4.142). The identity (4.143) can be proven
identically if one starts with the following Dodgson condensation identity:

(4.151) $\begin{split}&\mathscr{E}_{s}\left\\{\begin{matrix}\nu+2\\\
\nu+2\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}0&\nu+1&\nu+2\\\
0&\nu+1&\nu+2\end{matrix}\right\\}=\mathscr{E}_{s}\left\\{\begin{matrix}0&\nu+2\\\
0&\nu+2\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}\nu+1&\nu+2\\\
\nu+1&\nu+2\end{matrix}\right\\}-\mathscr{E}_{s}\left\\{\begin{matrix}0&\nu+2\\\
\nu+1&\nu+2\end{matrix}\right\\}\cdot\mathscr{E}_{s}\left\\{\begin{matrix}\nu+1&\nu+2\\\
0&\nu+2\end{matrix}\right\\}.\end{split}$

∎

######  Remark 4.11.

So far we have treated the offsets as arbitrary integers, however from the
pure $r,s$ recurrences of Theorem 4.4 it is clear that only those offsets that
are distinct modulo 3 are independent. Thus we can use any three consecutive
values of $r,s$. This is the analogue to the Toeplitz case where the
reflection or Verblunsky coefficients corresponded to the two offsets of $\pm
1$. This fact is reinforced by the results of Theorem 4.10 which show that the
moment determinants $D_{n}^{(0)},E_{n}^{(0)}$ can be constructed from
polynomial tails with $r=0,1,2$ and $s=0,-1,-2$ respectively.

##  5\. Multiple integral representations

The joint eigenvalue PDF of (1.9) is fundamental from our viewpoint and it is
from this that we construct various marginal distributions, otherwise known as
$n$-particle correlation functions, through integration over a set of
”internal” variables, the complement of $n$ ”external” variables. All aspects
of the theory will have representations of this form: the determinants, the
bi-orthogonal polynomials and the reproducing kernels. Thus our first result
of this nature is the $0$-point correlation or normalisation integral. This is
the extension of the Toeplitz result, Eq.(2.2), of [35].

######  Theorem 5.1.

The moment determinants, and therefore the normalisation of the bi-orthogonal
polynomials, possess the multiple integral representations

(5.1)
$D_{n}^{(r)}=\frac{1}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{2}}{2\pi\textrm{i}\zeta_{2}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r}_{j}\prod_{1\leq
j<k\leq n}(\zeta_{k}-\zeta_{j})(\zeta^{-2}_{k}-\zeta^{-2}_{j}),$

and

(5.2)
$E_{n}^{(s)}=\frac{1}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{2}}{2\pi\textrm{i}\zeta_{2}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-s}_{j}\prod_{1\leq
j<k\leq n}(\zeta^{2}_{k}-\zeta^{2}_{j})(\zeta^{-1}_{k}-\zeta^{-1}_{j}).$

###### Proof.

We have

(5.3) $\prod_{1\leq j<k\leq n}(\zeta_{k}-\zeta_{j})=\underset{1\leq j,k\leq
n}{\det}\\{\zeta^{k-1}_{j}\\}=\underset{1\leq j,k\leq
n}{\det}\\{P_{k-1}(\zeta_{j})\\},$

and

(5.4) $\prod_{1\leq j<k\leq n}(\zeta^{-2}_{k}-\zeta^{-2}_{j})=\underset{1\leq
j,k\leq n}{\det}\\{\zeta^{-2(k-1)}_{j}\\}=\underset{1\leq j,k\leq
n}{\det}\\{Q_{k-1}(\zeta^{-2}_{j})\\}.$

So

(5.5)
$\begin{split}&\frac{1}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{2}}{2\pi\textrm{i}\zeta_{2}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r}_{j}\prod_{1\leq
j<k\leq n}(\zeta_{k}-\zeta_{j})(\zeta^{-2}_{k}-\zeta^{-2}_{j})\\\
&=\frac{1}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{2}}{2\pi\textrm{i}\zeta_{2}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r}_{j}\underset{1\leq
j,k\leq n}{\det}\\{P_{k-1}(\zeta_{j})\\}\underset{1\leq j,k\leq
n}{\det}\\{Q_{k-1}(\zeta^{-2}_{j})\\}\\\
&=\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{2}}{2\pi\textrm{i}\zeta_{2}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r}_{j}\underset{1\leq
j,k\leq
n}{\det}\\{P_{k-1}(\zeta_{j})\\}\prod_{j=1}^{n}Q_{j-1}(\zeta^{-2}_{j})\\\
&=\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{2}}{2\pi\textrm{i}\zeta_{2}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r}_{j}\underset{1\leq
j,k\leq n}{\det}\\{P_{k-1}(\zeta_{j})Q_{j-1}(\zeta^{-2}_{j})\\}\\\
&=\underset{1\leq j,k\leq
n}{\det}\left\\{\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta}{2\pi\textrm{i}\zeta}w(\zeta)\zeta^{-r}P_{k-1}(\zeta)Q_{j-1}(\zeta^{-2})\right\\}=\underset{0\leq
j,k\leq
n-1}{\det}\left\\{h^{(r)}_{j}\delta_{j,k}\right\\}=\prod_{j=0}^{n-1}h^{(r)}_{j}=D^{(r)}_{n},\end{split}$

where we have used (2.16) and (3.3). The formula (5.2) can be established
similarly using (2.17) and (3.9). ∎

So, in particular we obtain (2.14) and (2.15) in view of (2.12), (2.13), (5.1)
and (5.2). The relationship (3.11) can be also found using the multiple
integral formulae above. To that end notice that

(5.6)
$(\zeta_{k}-\zeta_{j})(\zeta^{-2}_{k}-\zeta^{-2}_{j})=\frac{1}{\zeta_{k}\zeta_{j}}(\zeta^{-1}_{k}-\zeta^{-1}_{j})(\zeta^{2}_{k}-\zeta^{2}_{j}).$

Therefore

(5.7) $\prod_{1\leq j<k\leq
n}(\zeta_{k}-\zeta_{j})(\zeta^{-2}_{k}-\zeta^{-2}_{j})=\psi\prod_{1\leq
j<k\leq n}(\zeta^{-1}_{k}-\zeta^{-1}_{j})(\zeta^{2}_{k}-\zeta^{2}_{j}),$

where

(5.8) $\psi=\prod_{1\leq j<k\leq
n}\frac{1}{\zeta_{k}\zeta_{j}}=\prod_{j=1}^{n}\zeta^{1-n}_{j},$

because

(5.9) $\psi^{2}=\prod_{1\leq j\neq k\leq
n}\frac{1}{\zeta_{k}\zeta_{j}}=\frac{\displaystyle\prod_{1\leq j,k\leq
n}\frac{1}{\zeta_{k}\zeta_{j}}}{\displaystyle\prod_{1\leq j=k\leq
n}\frac{1}{\zeta_{k}\zeta_{j}}}=\prod_{j=1}^{n}\zeta^{2}_{j}\prod_{1\leq
j,k\leq n}\zeta^{-1}_{k}\zeta^{-1}_{j}=\prod_{j=1}^{n}\zeta^{2-2n}_{j}.$

Note that

(5.10)
$\begin{split}\mathcal{D}_{n}[w(\zeta)\zeta^{-r}]&=\frac{1}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{2}}{2\pi\textrm{i}\zeta_{2}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r}_{j}\prod_{1\leq
j<k\leq n}(\zeta_{k}-\zeta_{j})(\zeta^{-2}_{k}-\zeta^{-2}_{j})\\\
&=\frac{1}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{2}}{2\pi\textrm{i}\zeta_{2}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{1-n-r}_{j}\prod_{1\leq
j<k\leq n}(\zeta^{-1}_{k}-\zeta^{-1}_{j})(\zeta^{2}_{k}-\zeta^{2}_{j})\\\
&=\mathcal{E}_{n}[w(\zeta)\zeta^{-(n+r-1)}],\end{split}$

which is equivalent to (3.11). More generally we have the following result.

######  Theorem 5.2.

For any integrable function $f$ we have:

(5.11)
$\mathcal{D}_{n}[w(\zeta)\zeta^{-r}f(\zeta)]=\mathcal{E}_{n}[w(\zeta)\zeta^{-n-r+1}f(\zeta)],$

or equivalently

(5.12)
$\mathcal{E}_{n}[w(\zeta)\zeta^{-s}f(\zeta)]=\mathcal{D}_{n}[w(\zeta)\zeta^{-s-1+n}f(\zeta)].$

###  5.1. Multiple integral formulae for the bi-orthogonal polynomials

It is well known that the bi-orthogonal polynomials in the Toeplitz system can
be written as averages of characteristic polynomials over the unitary group,
see Eq.(2.16,18) of [35]. In the following Theorem we give the analogous
results for the $2j-k$ and $j-2k$ polynomials. Another interpretation of such
integrals is as rational modifications of the weight, known as Christoffel-
Darboux-Uvarov transformations of the underlying bi-orthogonal system in the
Toeplitz case, which was exhaustively treated in [72].

######  Theorem 5.3.

We have the following multiple integral representations for $2j-k$ and $j-2k$
polynomials:

(5.13) $\displaystyle P_{n}(z;r)$
$\displaystyle=\frac{1}{D^{(r)}_{n}}\mathcal{D}_{n}[w(\zeta)\zeta^{-r}(z-\zeta)],$
(5.14) $\displaystyle Q_{n}(z;r)$
$\displaystyle=\frac{1}{D^{(r)}_{n}}\mathcal{D}_{n}[w(\zeta)\zeta^{-r}(z-\zeta^{-2})],$

(5.15) $\displaystyle R_{n}(z;s)$
$\displaystyle=\frac{1}{E^{(s)}_{n}}\mathcal{E}_{n}[w(\zeta)\zeta^{-s}(z-\zeta^{2})],$
(5.16) $\displaystyle S_{n}(z;s)$
$\displaystyle=\frac{1}{E^{(s)}_{n}}\mathcal{E}_{n}[w(\zeta)\zeta^{-s}(z-\zeta^{-1})].$

###### Proof.

We only prove the formula for $P_{n}$ as the other ones can be found
similarly.

(5.17) $\begin{split}&\mathcal{D}_{n}[w(\zeta)\zeta^{-r}(z-\zeta)]\\\
&=\frac{1}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r}_{j}\prod_{j=1}^{n}(z-\zeta_{j})\prod_{1\leq
j<k\leq n}(\zeta_{k}-\zeta_{j})(\zeta^{-2}_{k}-\zeta^{-2}_{j})\\\
&=\frac{(-1)^{n}}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r}_{j}\prod_{0\leq
j<k\leq n}(\zeta_{k}-\zeta_{j})\prod_{1\leq j<k\leq
n}(\zeta^{-2}_{k}-\zeta^{-2}_{j}),\end{split}$

where we have denoted $z$ by $\zeta_{0}$. Therefore

(5.18) $\begin{split}&\mathcal{D}_{n}[w(\zeta)\zeta^{-r}(z-\zeta)]\\\
&=\frac{(-1)^{n}}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r}_{j}\underset{\begin{subarray}{c}1\leq
j\leq n\\\ 1\leq k\leq
n+1\end{subarray}}{\det}\begin{pmatrix}P_{k-1}(\zeta_{0})\\\
P_{k-1}(\zeta_{j})\end{pmatrix}\underset{\begin{subarray}{c}1\leq j\leq n\\\
1\leq k\leq n\end{subarray}}{\det}\\{Q_{k-1}(\zeta^{-2}_{j})\\}\\\
&=(-1)^{n}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r}_{j}\underset{\begin{subarray}{c}1\leq
j\leq n\\\ 1\leq k\leq
n+1\end{subarray}}{\det}\begin{pmatrix}P_{k-1}(\zeta_{0})\\\
P_{k-1}(\zeta_{j})\end{pmatrix}\prod_{j=1}^{n}Q_{j-1}(\zeta^{-2}_{j})\\\
&=(-1)^{n}\underset{\begin{subarray}{c}1\leq j\leq n\\\ 1\leq k\leq
n+1\end{subarray}}{\det}\begin{pmatrix}P_{k-1}(\zeta_{0})\\\
\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta}{2\pi\textrm{i}\zeta}w(\zeta)\zeta^{-r}P_{k-1}(\zeta)Q_{j-1}(\zeta^{-2})\end{pmatrix}\\\
&=(-1)^{n}\det\begin{pmatrix}P_{0}(z)&\cdots&P_{n-1}(z)&P_{n}(z)\\\
h^{(r)}_{0}&\cdots&0&0\\\ \vdots&\ddots&\vdots&\vdots\\\
0&\cdots&h^{(r)}_{n-1}&0\end{pmatrix}=P_{n}(z;r)\prod_{j=0}^{n-1}h^{(r)}_{j}=D^{(r)}_{n}P_{n}(z;r).\end{split}$

∎

###  5.2. Multiple integral formulae for the reproducing kernels

We will require the following important lemma, which is known as the
”integration over successive variables”, in a subsequent result. This is the
extension of the result in the Toeplitz case, which is easy to derive
following the line of Proposition 5.1.2 of [30].

######  Lemma 5.4 (Gaudin’s Theorem).

The integration over the final variable $\zeta_{m}$ of the $m\times m$
determinant constructed from reproducing kernel entries has the evaluation

(5.19)
$\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{m}}{2\pi\textrm{i}\zeta_{m}}w(\zeta_{m})\zeta^{-r}_{m}\underset{\begin{subarray}{c}1\leq
j\leq m\\\ 1\leq k\leq
m\end{subarray}}{\det}\left[K_{n}(\zeta^{-2}_{k},\zeta_{j};r)\right]=(n-m+2)\underset{\begin{subarray}{c}1\leq
j\leq m-1\\\ 1\leq k\leq
m-1\end{subarray}}{\det}\left[K_{n}(\zeta^{-2}_{k},\zeta_{j};r)\right],$

and

(5.20)
$\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{m}}{2\pi\textrm{i}\zeta_{m}}w(\zeta_{m})\zeta^{-s}_{m}\underset{\begin{subarray}{c}1\leq
j\leq m\\\ 1\leq k\leq
m\end{subarray}}{\det}\left[L_{n}(\zeta^{-1}_{k},\zeta^{2}_{j};s)\right]=(n-m+2)\underset{\begin{subarray}{c}1\leq
j\leq m-1\\\ 1\leq k\leq
m-1\end{subarray}}{\det}\left[L_{n}(\zeta^{-1}_{k},\zeta^{2}_{j};s)\right].$

###### Proof.

Let us use $j$ and $k$ respectively as indices of rows and columns. Expanding
the determinant in the integrand along the last row gives

(5.21) $\underset{\begin{subarray}{c}1\leq j\leq m\\\ 1\leq k\leq
m\end{subarray}}{\det}\left[K_{n}(\zeta^{-2}_{k},\zeta_{j};r)\right]=\sum^{m-1}_{k=1}(-1)^{m+k}K_{n}(\zeta^{-2}_{k},\zeta_{m};r)\underset{\begin{subarray}{c}1\leq
j\leq m-1\\\ 1\leq\ell\leq m\\\ \ell\neq
k\end{subarray}}{\det}\left[K_{n}(\zeta^{-2}_{\ell},\zeta_{j};r)\right]+K_{n}(\zeta^{-2}_{m},\zeta_{m};r)\underset{\begin{subarray}{c}1\leq
j\leq m-1\\\ 1\leq k\leq
m-1\end{subarray}}{\det}\left[K_{n}(\zeta^{-2}_{k},\zeta_{j};r)\right]$ (5.22)
$=\sum^{m-1}_{k=1}(-1)^{m+k}\underset{\begin{subarray}{c}1\leq j\leq m-1\\\
1\leq\ell\leq m-1\\\ \ell\neq
k\end{subarray}}{\det}\left[K_{n}(\zeta^{-2}_{\ell},\zeta_{j};r),K_{n}(\zeta^{-2}_{k},\zeta_{m};r)K_{n}(\zeta^{-2}_{m},\zeta_{j};r)\right]+K_{n}(\zeta^{-2}_{m},\zeta_{m};r)\underset{\begin{subarray}{c}1\leq
j\leq m-1\\\ 1\leq k\leq
m-1\end{subarray}}{\det}\left[K_{n}(\zeta^{-2}_{k},\zeta_{j};r)\right].$

Recalling the normalisation (3.42) and the projection relations (3.43) for the
reproducing kernel we deduce

(5.23)
$\begin{split}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{m}}{2\pi\textrm{i}\zeta_{m}}w(\zeta_{m})\zeta^{-r}_{m}\underset{\begin{subarray}{c}1\leq
j\leq m\\\ 1\leq k\leq
m\end{subarray}}{\det}\left[K_{n}(\zeta^{-2}_{k},\zeta_{j};r)\right]&=\sum^{m-1}_{k=1}(-1)^{m+k}\underset{\begin{subarray}{c}1\leq
j\leq m-1\\\ 1\leq\ell\leq m-1\\\ \ell\neq
k\end{subarray}}{\det}\left[K_{n}(\zeta^{-2}_{\ell},\zeta_{j};r),K_{n}(\zeta^{-2}_{k},\zeta_{j};r)\right]\\\
&+(n+1)\underset{\begin{subarray}{c}1\leq j\leq m-1\\\ 1\leq k\leq
m-1\end{subarray}}{\det}\left[K_{n}(\zeta^{-2}_{k},\zeta_{j};r)\right],\end{split}$

By $m-k-1$ successive swaps of adjacent columns we have

$\underset{\begin{subarray}{c}1\leq j\leq m-1\\\ 1\leq\ell\leq m-1\\\ \ell\neq
k\end{subarray}}{\det}\left[K_{n}(\zeta^{-2}_{\ell},\zeta_{j};r),K_{n}(\zeta^{-2}_{k},\zeta_{j};r)\right]=(-1)^{m-k-1}\underset{\begin{subarray}{c}1\leq
j\leq m-1\\\ 1\leq\ell\leq
m-1\end{subarray}}{\det}\left[K_{n}(\zeta^{-2}_{\ell},\zeta_{j};r)\right].$

Equation (5.19) now follows by combining the last two equations. The proof of
(5.20) is similar and we do not provide the details here. ∎

######  Theorem 5.5.

The reproducing kernels for $2j-k$ and $j-2k$ systems have the following
multiple integral representations:

(5.24)
$\begin{split}K_{n}(z_{2},z_{1};r)&=\frac{1}{D^{(r)}_{n+1}}\mathcal{D}_{n}[w(\zeta)\zeta^{-r}(z_{1}-\zeta)(z_{2}-\zeta^{-2})],\end{split}$

and

(5.25)
$L_{n}(z_{2},z_{1};s)=\frac{1}{E^{(s)}_{n+1}}\mathcal{E}_{n}[w(\zeta)\zeta^{-s}(z_{1}-\zeta^{2})(z_{2}-\zeta^{-1})]=\left(z_{1}z_{2}\right)^{n}K_{n}(z_{1}^{-1},z_{2}^{-1};s-n).$

###### Proof.

For simplicity of notation, in this proof we denote $P_{n}(z;r),Q_{n}(z;r)$
and $K_{n}(z_{2},z_{1};r)$ respectively by $P_{n}(z),Q_{n}(z)$ and
$K_{n}(z_{2},z_{1})$. We have

$\begin{split}&\mathcal{D}_{n}[w(\zeta)\zeta^{-r}(\zeta-
z_{1})(\zeta^{-2}-z_{2})]\\\
&=\frac{1}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r}_{j}\prod_{j=1}^{n}(\zeta_{j}-z_{1})(\zeta_{j}^{-2}-z_{2})\prod_{1\leq
j<k\leq n}(\zeta_{k}-\zeta_{j})(\zeta^{-2}_{k}-\zeta^{-2}_{j})\\\
&=\frac{1}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r}_{j}\prod_{0\leq
j<k\leq n}(\zeta_{k}-\zeta_{j})(\zeta^{-2}_{k}-\zeta^{-2}_{j}),\end{split}$

where we have temporarily denoted $z_{1}$ and $z_{2}$ respectively by
$\zeta_{0}$ and $\zeta^{-2}_{0}$. Therefore

$\begin{split}&\mathcal{D}_{n}[w(\zeta)\zeta^{-r}(\zeta-
z_{1})(\zeta^{-2}-z_{2})]\\\
&=\frac{1}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r}_{j}\underset{\begin{subarray}{c}1\leq
j\leq n\\\ 1\leq k\leq
n+1\end{subarray}}{\det}\begin{pmatrix}P_{k-1}(\zeta_{0})\\\
P_{k-1}(\zeta_{j})\end{pmatrix}\underset{\begin{subarray}{c}1\leq j\leq n\\\
1\leq k\leq n+1\end{subarray}}{\det}\begin{pmatrix}Q_{k-1}(\zeta^{-2}_{0})\\\
Q_{k-1}(\zeta^{-2}_{j})\end{pmatrix}\\\
&=\frac{1}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r}_{j}\underset{\begin{subarray}{c}1\leq
j\leq n\\\ 1\leq k\leq
n+1\end{subarray}}{\det}\begin{pmatrix}P_{k-1}(\zeta_{0})\\\
P_{k-1}(\zeta_{j})\end{pmatrix}\underset{\begin{subarray}{c}1\leq j\leq n\\\
1\leq k\leq n+1\end{subarray}}{\det}\begin{pmatrix}Q_{k-1}(\zeta^{-2}_{0})\\\
Q_{k-1}(\zeta^{-2}_{j})\end{pmatrix}^{T}\\\
&=\frac{D^{(r)}_{n+1}}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r}_{j}\det\begin{pmatrix}P_{0}(\zeta_{0})&\cdots&P_{n}(\zeta_{0})\\\
\vdots&\vdots&\vdots\\\ P_{0}(\zeta_{n})&\cdots&P_{n}(\zeta_{n})\\\
\end{pmatrix}\det\begin{pmatrix}\frac{Q_{0}(\zeta^{-2}_{0})}{h^{(r)}_{0}}&\cdots&\frac{Q_{0}(\zeta^{-2}_{n})}{h^{(r)}_{0}}\\\
\vdots&\vdots&\vdots\\\
\frac{Q_{n}(\zeta^{-2}_{0})}{h^{(r)}_{n}}&\cdots&\frac{Q_{n}(\zeta^{-2}_{n})}{h^{(r)}_{n}}\\\
\end{pmatrix}\\\
&=\frac{D^{(r)}_{n+1}}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r}_{j}\underset{\begin{subarray}{c}1\leq
j\leq n\\\ 1\leq k\leq
n\end{subarray}}{\det}\begin{pmatrix}\displaystyle\sum_{\nu=0}^{n}\frac{P_{\nu}(\zeta_{0})Q_{\nu}(\zeta^{-2}_{0})}{h^{(r)}_{\nu}}&\displaystyle\sum_{\nu=0}^{n}\frac{P_{\nu}(\zeta_{0})Q_{\nu}(\zeta^{-2}_{k})}{h^{(r)}_{\nu}}\\\
\displaystyle\sum_{\nu=0}^{n}\frac{P_{\nu}(\zeta_{j})Q_{\nu}(\zeta^{-2}_{0})}{h^{(r)}_{\nu}}&\displaystyle\sum_{\nu=0}^{n}\frac{P_{\nu}(\zeta_{j})Q_{\nu}(\zeta^{-2}_{k})}{h^{(r)}_{\nu}}\end{pmatrix}\\\
&=\frac{D^{(r)}_{n+1}}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r}_{j}\underset{\begin{subarray}{c}1\leq
j\leq n\\\ 1\leq k\leq
n\end{subarray}}{\det}\begin{pmatrix}K_{n}(\zeta^{-2}_{0},\zeta_{0})&K_{n}(\zeta^{-2}_{k},\zeta_{0})\\\
K_{n}(\zeta^{-2}_{0},\zeta_{j})&K_{n}(\zeta^{-2}_{k},\zeta_{j})\end{pmatrix}.\end{split}$

Now we apply Theorem 5.4 :

(5.26) $\begin{split}&\mathcal{D}_{n}[w(\zeta)\zeta^{-r}(\zeta-
z_{1})(\zeta^{-2}-z_{2})]=(n+2-(n+1))\\\
&\times\frac{D^{(r)}_{n+1}}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n-1}}{2\pi\textrm{i}\zeta_{n-1}}\prod_{j=1}^{n-1}w(\zeta_{j})\zeta^{-r}_{j}\underset{\begin{subarray}{c}1\leq
j\leq n-1\\\ 1\leq k\leq
n-1\end{subarray}}{\det}\begin{pmatrix}K_{n}(\zeta^{-2}_{0},\zeta_{0})&K_{n}(\zeta^{-2}_{k},\zeta_{0})\\\
K_{n}(\zeta^{-2}_{0},\zeta_{j})&K_{n}(\zeta^{-2}_{k},\zeta_{j})\end{pmatrix}.\end{split}$

Repeating this $n-1$ times yields

(5.27) $\mathcal{D}_{n}[w(\zeta)\zeta^{-r}(\zeta-
z_{1})(\zeta^{-2}-z_{2})]=\prod_{j=2}^{n+1}(n+2-j)\frac{D^{(r)}_{n+1}}{n!}K_{n}(\zeta^{-2}_{0},\zeta_{0}),$

and thus $\mathcal{D}_{n}[w(\zeta)\zeta^{-r}(\zeta-
z_{1})(\zeta^{-2}-z_{2})]=D^{(r)}_{n+1}K_{n}(z_{2},z_{1})$. The equation
(5.25) can be established similarly and we do not provide the details here. ∎

######  Remark 5.6.

One can also see the consistency of multiple integral formulae for the
reproducing kernels with the definitions (3.36) and (3.37) by considering the
leading asymptotic behaviour as $z$ or $\mathcal{z}$ tend to infinity. For
example as $z\to\infty$ we have

$K_{n}(z,\mathcal{z};r):=\sum_{j=0}^{n}\frac{1}{h^{(r)}_{j}}Q_{j}(z;r)P_{j}(\mathcal{z};r)=\frac{z^{n}}{h^{(r)}_{n}}P_{j}(\mathcal{z};r)\left(1+\mathcal{O}\left(\frac{1}{z}\right)\right),\qquad
z\to\infty,$

and, expectedly, from the right hand side of (5.24) we find the same
asymptotic behaviour:

$\frac{1}{D^{(r)}_{n+1}}\mathcal{D}_{n}[w(\zeta)\zeta^{-r}(\mathcal{z}-\zeta)(z-\zeta^{-2})]=\frac{1}{D^{(r)}_{n+1}}\mathcal{D}_{n}[w(\zeta)\zeta^{-r}(\mathcal{z}-\zeta)z]\left(1+\mathcal{O}\left(\frac{1}{z}\right)\right)$
$\hskip
22.76219pt=\frac{z^{n}}{h^{(r)}_{n}D^{(r)}_{n}}\mathcal{D}_{n}[w(\zeta)\zeta^{-r}(\mathcal{z}-\zeta)]\left(1+\mathcal{O}\left(\frac{1}{z}\right)\right)=\frac{z^{n}}{h^{(r)}_{n}}P_{j}(\mathcal{z};r)\left(1+\mathcal{O}\left(\frac{1}{z}\right)\right),\qquad
z\to\infty,$

by (5.13). Similar consistency checks can be done for $K_{n}(z,\mathcal{z};r)$
as $\mathcal{z}\to\infty$, and also for $L_{n}(z,\mathcal{z};r)$ as
$z\to\infty$ and separately as $\mathcal{z}\to\infty$.

######  Theorem 5.7.

We have

(5.28) $\displaystyle P_{n}(z;r)$
$\displaystyle=P_{n}(z;r-2)+\frac{h^{(r-2)}_{n}}{Q_{n}(0;r-2)}\sum_{\ell=0}^{n-1}\frac{Q_{\ell}(0;r-2)}{h^{(r-2)}_{\ell}}P_{\ell}(z;r-2),$
(5.29) $\displaystyle Q_{n}(z;r)$
$\displaystyle=Q_{n}(z;r+1)+\frac{h^{(r+1)}_{n}}{P_{n}(0;r+1)}\sum_{\ell=0}^{n-1}\frac{P_{\ell}(0;r+1)}{h^{(r+1)}_{\ell}}Q_{\ell}(z;r+1),$
(5.30) $\displaystyle R_{n}(z;s)$
$\displaystyle=R_{n}(z;s-1)+\frac{g^{(s-1)}_{n}}{S_{n}(0;s-1)}\sum_{\ell=0}^{n-1}\frac{S_{\ell}(0;s-1)}{g^{(s-1)}_{\ell}}R_{\ell}(z;s-1),$
(5.31) $\displaystyle S_{n}(z;s)$
$\displaystyle=S_{n}(z;s+2)+\frac{g^{(s+2)}_{n}}{R_{n}(0;s+2)}\sum_{\ell=0}^{n-1}\frac{R_{\ell}(0;s+2)}{g^{(s+2)}_{\ell}}S_{\ell}(z;s+2).$

###### Proof.

These identities can be proven if one considers special arguments for the
reproducing kernel. Indeed, to prove (5.28) we consider

(5.32)
$K_{n}(0,z;r)=\sum_{\ell=0}^{n}\frac{1}{h^{(r)}_{\ell}}Q_{\ell}(0;r)P_{\ell}(z;r),$

by (3.36). On the other hand from (5.24) we have

(5.33)
$K_{n}(0,z;r)=\frac{(-1)^{n}}{D^{(r)}_{n+1}}\mathcal{D}_{n}[w(\zeta)\zeta^{-r-2}(z-\zeta)]=(-1)^{n}\frac{D^{(r+2)}_{n}}{D^{(r)}_{n+1}}P_{n}(z;r+2),$

where the last equality follows from (5.13). From (4.135) we have

(5.34)
$\frac{Q_{n}(0;r)}{h^{(r)}_{n}}=(-1)^{n}\frac{D^{(r+2)}_{n}}{D^{(r)}_{n+1}}.$

Now (5.28) follows from (5.32), (5.33), and (5.34). The identity (5.29) can be
proven by considering:

(5.35)
$K_{n}(z,0;r)=\sum_{\ell=0}^{n}\frac{1}{h^{(r)}_{\ell}}Q_{\ell}(z;r)P_{\ell}(0;r),$

and

(5.36)
$K_{n}(z,0;r)=\frac{(-1)^{n}}{D^{(r)}_{n+1}}\mathcal{D}_{n}[w(\zeta)\zeta^{-r+1}(z-\zeta^{-2})]=(-1)^{n}\frac{D^{(r-1)}_{n}}{D^{(r)}_{n+1}}Q_{n}(z;r-1),$

by (5.24) and (5.14). Equation (5.29) now follows from (5.35), (5.36) and

(5.37)
$\frac{P_{n}(0;r)}{h^{(r)}_{n}}=(-1)^{n}\frac{D^{(r-1)}_{n}}{D^{(r)}_{n+1}},$

which is obtained by (4.134). (5.30) and (5.31) can be proven similarly,
respectively by considering $L_{n}(0,z;s)$ and $L_{n}(z,0;s)$ and employing

(5.38)
$\frac{S_{n}(0;s)}{g^{(s)}_{n}}=(-1)^{n}\frac{E^{(s+1)}_{n}}{E^{(s)}_{n+1}},\quad\text{and}\quad\frac{R_{n}(0;s)}{g^{(s)}_{n}}=(-1)^{n}\frac{E^{(s-2)}_{n}}{E^{(s)}_{n+1}},$

which are consequences of (4.136) and (4.137) respectively. ∎

######  Corollary 5.7.1.

The following mixed recurrence relations hold

(5.39) $\displaystyle P_{n}(z;r)$
$\displaystyle=P_{n}(z;r+2)+\frac{h^{(r)}_{n}}{h^{(r+2)}_{n-1}}P_{n-1}(z;r+2),$
(5.40) $\displaystyle Q_{n}(z;r)$
$\displaystyle=Q_{n}(z;r-1)+\frac{h^{(r)}_{n}}{h^{(r-1)}_{n-1}}Q_{n-1}(z;r-1),$

(5.41) $\displaystyle R_{n}(z;s)$
$\displaystyle=R_{n}(z;s+1)+\frac{g^{(s)}_{n}}{g^{(s+1)}_{n-1}}R_{n-1}(z;s+1),$
(5.42) $\displaystyle S_{n}(z;s)$
$\displaystyle=S_{n}(z;s-2)+\frac{g^{(s)}_{n}}{g^{(s-2)}_{n-1}}S_{n-1}(z;s-2).$

###### Proof.

Writing (5.28) with the replacement $r\mapsto r+2$ for $n$ and $n-1$ and
subtracting gives

$\frac{Q_{n}(0;r)}{h^{(r)}_{n}}P_{n}(z;r+2)-\frac{Q_{n-1}(0;r)}{h^{(r)}_{n-1}}P_{n-1}(z;r+2)=\frac{1}{h^{(r)}_{n}}Q_{n}(0;r)P_{n}(z;r),$

or equivalently

$P_{n}(z;r)=P_{n}(z;r+2)-\frac{h^{(r)}_{n}}{Q_{n}(0;r)}\frac{Q_{n-1}(0;r)}{h^{(r)}_{n-1}}P_{n-1}(z;r+2).$

Now (5.39) immediately follows by recalling (5.34) and (3.3). The other three
identities can be shown in a similar fashion. ∎

For fixed offset values $\hat{r},\hat{s}\in{\mathbb{Z}}$, let us consider the
following $2j-k$ and $j-2k$ polynomial bases:

(5.43)
$\mathscr{P}^{(\hat{r})}_{n}:=\\{P_{m}(z;\hat{r})\\}^{n}_{m=0},\qquad\mathscr{Q}^{(\hat{r})}_{n}:=\\{Q_{m}(z;\hat{r})\\}^{n}_{m=0},\qquad\mathscr{R}^{(\hat{s})}_{n}:=\\{R_{m}(z;\hat{s})\\}^{n}_{m=0},\qquad\mathscr{S}^{(\hat{s})}_{n}:=\\{S_{m}(z;\hat{s})\\}^{n}_{m=0}.$

We obviously assume that the corresponding determinants do not vanish so that
the polynomials exist (and are unique, see Theorems 3.1 and 3.2). The
following theorem characterizes the number of polynomials needed to represent
$P_{n}(z,r),\ Q_{n}(z,r),\ R_{n}(z,s)$ and $S_{n}(z,s)$ respectively in terms
of polynomials in the bases $\mathscr{P}^{(\hat{r})}_{n}$,
$\mathscr{Q}^{(\hat{r})}_{n}$, $\mathscr{R}^{(\hat{s})}_{n}$, and
$\mathscr{S}^{(\hat{s})}_{n}$ for various values of $\hat{r}$ and $\hat{s}$.

######  Theorem 5.8.

Assume the non-vanishing requirements of Theorems 3.1 and 3.2 so that the
polynomial bases $\mathscr{P}^{(\hat{r})}_{n}$, $\mathscr{Q}^{(\hat{r})}_{n}$,
$\mathscr{R}^{(\hat{s})}_{n}$, and $\mathscr{S}^{(\hat{s})}_{n}$ are well-
defined as in (5.43) and let $m\in{\mathbb{N}}$. Then

  * 1.

If it is desired to express $P_{n}(z;r)$ as a linear combination of
polynomials in the basis $\mathscr{P}^{(r-2m)}_{n}$, then one needs to use all
$n+1$ polynomials in $\mathscr{P}^{(r-2m)}_{n}$. While if it is desired to
express $P_{n}(z;r)$ in terms of polynomials in the basis
$\mathscr{P}^{(r+2m)}_{n}$, then one needs to only use the following $m+1$
polynomials in the subset

$\left\\{P_{n}(z;r+2m),P_{n-1}(z;r+2m),\cdots,P_{n-m}(z;r+2m)\right\\}\subset\mathscr{P}^{(r+2m)}_{n}.$

  * 2.

If it is desired to express $zP_{n}(z;r)$ as a linear combination of
polynomials in $\mathscr{P}^{(r-2m+1)}_{n+1}$, then one needs to use all $n+2$
polynomials in $\mathscr{P}^{(r-2m+1)}_{n+1}$. While if it is desired to
express $zP_{n}(z;r)$ in terms of polynomials in
$\mathscr{P}^{(r+2m-1)}_{n+1}$, then one needs to only use the $m+1$
polynomials in the subset

$\\{P_{n+1}(z;r+2m-1),P_{n}(z;r+2m-1),\cdots,P_{n+1-m}(z;r+2m-1)\\}\subset\mathscr{P}^{(r+2m-1)}_{n+1}.$

  * 3.

If it is desired to express $Q_{n}(z;r)$ as a linear combination of
polynomials in $\mathscr{Q}^{(r+m)}_{n}$, then one needs to use all $n+1$
polynomials in $\mathscr{Q}^{(r+m)}_{n}$. While if it is desired to express
$Q_{n}(z;r)$ in terms of polynomials in $\mathscr{Q}^{(r-m)}_{n}$, then one
needs to only use the $m+1$ polynomials in the subset

$\\{Q_{n}(z;r-m),Q_{n-1}(z;r-m),\cdots,Q_{n-m}(z;r-m)\\}\subset\mathscr{Q}^{(r-m)}_{n}.$

  * 4.

If it is desired to express $R_{n}(z;s)$ as a linear combination of
polynomials in $\mathscr{R}^{(s-m)}_{n}$, then one needs to use all $n+1$
polynomials in $\mathscr{R}^{(s-m)}_{n}$. While if it is desired to express
$R_{n}(z;s)$ in terms of polynomials in $\mathscr{R}^{(s+m)}_{n}$, then one
needs to only use the $m+1$ polynomials in the subset

$\\{R_{n}(z;s+m),R_{n-1}(z;s+m),\cdots,R_{n-m}(z;s+m)\\}\subset\mathscr{R}^{(s+m)}_{n}.$

  * 5.

If it is desired to express $S_{n}(z;s)$ as a linear combination of
polynomials in $\mathscr{S}^{(s+2m)}_{n}$, then one needs to use all $n+1$
polynomials in $\mathscr{S}^{(s+2m)}_{n}$. While if it is desired to express
$S_{n}(z;s)$ in terms of polynomials in $\mathscr{S}^{(s-2m)}_{n}$, then one
needs to only use the $m+1$ polynomials in the subset

$\\{S_{n}(z;s-2m),S_{n-1}(z;s-2m),\cdots,S_{n-m}(z;s-2m)\\}\subset\mathscr{S}^{(s-2m)}_{n}.$

  * 6.

If it is desired to express $zS_{n}(z;s)$ as a linear combination of
polynomials in $\mathscr{S}^{(s+2m-1)}_{n+1}$, then one needs to use all $n+2$
polynomials in $\mathscr{S}^{(s+2m-1)}_{n+1}$. While if it is desired to
express $zS_{n}(z;s)$ in terms of polynomials in
$\mathscr{P}^{(s-2m+1)}_{n+1}$, then one needs to only use the $m+1$
polynomials in the subset

$\\{S_{n+1}(z;s-2m+1),S_{n}(z;s-2m+1),\cdots,S_{n+1-m}(z;s-2m+1)\\}\subset\mathscr{S}^{(s-2m+1)}_{n+1}.$

###### Proof.

Theorem 5.7 establishes the statement of this theorem for expressing
$P_{n}(z,r),\ Q_{n}(z,r),\ R_{n}(z,s)$ and $S_{n}(z,s)$ respectively in terms
of polynomials in $\mathscr{P}^{(r-2)}_{n}$, $\mathscr{Q}^{(r+1)}_{n}$,
$\mathscr{R}^{(s-1)}_{n}$, and $\mathscr{S}^{(s+2)}_{n}$, and the Corollary
5.7.1 establishes this for expressing $P_{n}(z,r),\ Q_{n}(z,r),\ R_{n}(z,s)$
and $S_{n}(z,s)$ respectively in terms of polynomials in
$\mathscr{P}^{(r+2)}_{n}$, $\mathscr{Q}^{(r-1)}_{n}$,
$\mathscr{R}^{(s+1)}_{n}$, and $\mathscr{S}^{(s-2)}_{n}$. The proof for
general $m$ follows from iterations of the aforementioned results of Theorem
5.7 and Corollary 5.7.1, and the non-vanishing determinant requirements of
Theorems 3.1 and 3.2 ensure that the coefficients involved all exist and are
non-zero. Let us now prove the first item for expressing $P_{n}(z;r)$ in the
basis $\mathscr{P}^{(r-2m)}_{n}$ via induction. The base of the induction for
$m=1$ clearly holds because of (5.28). Passing from the induction hypothesis
to the desired result is obvious because by (5.28) we have
$P_{\ell}(z;r-2(m-1))=\sum_{\nu=0}^{\ell}\hat{A}_{\nu,\ell}P_{\nu}(z;r-2m)$,
where the constants $\hat{A}_{\nu,\ell}$ are also described in (5.28). Thus
the induction hypothesis
$P_{n}(z;r)=\sum_{\ell=0}^{n}A_{m,n}P_{\ell}(z;r-2(m-1))$, immediately implies
the desired result. To prove the first item for expressing $P_{n}(z;r)$ in the
basis $\mathscr{P}^{(r+2m)}_{n}$ we use induction again. The base of the
induction clearly holds due to (5.39). Now the induction hypothesis

(5.44) $P_{n}(z;r)=\sum_{\ell=0}^{m-1}B_{m-1,n}P_{n-\ell}(z;r+2(m-1)),$

in conjunction with

(5.45)
$P_{n-\ell}(z;r+2(m-1))=P_{n-\ell}(z;r+2m)+\frac{h^{(r+2(m-1))}_{n-\ell}}{h^{(r+2m)}_{n-\ell-1}}P_{n-\ell-1}(z;r+2m),$

yields the desired result. The proofs of the other items follow similarly,
where the only extra needed relations are

(5.46) $\displaystyle zP_{n}(z;r)$
$\displaystyle=P_{n+1}(z;r+1)-\delta_{n}^{(r+1)}P_{n}(z;r+1),$

(5.47) $\displaystyle zS_{n}(z;s)$
$\displaystyle=S_{n+1}(z;s-1)-\gamma_{n}^{(s-1)}S_{n}(z;s-1).$

which are obtained from (4.11) and (4.63) respectively. ∎

In the Toeplitz case there exist relations which allow one of the polynomials
to be expressed in terms of a linear combination of two of the other one, see
Eq.(2.21,22) of [35]. These two relations are also the components of the first
order recurrence, Eq.(2.70), in the matrix variable, Eq.(2.69). Furthermore
these two relations are somewhat symmetrical. However for the present system
we find the following single, unsymmetrical relation, of $Q_{n}$ in terms of a
bilinear combination of the $P_{n}$, and the analogous relation for the
$R_{n},S_{n}$ system.

######  Theorem 5.9.

It holds that

(5.48)
$Q_{n}(z^{2};r)=\frac{D^{(r+2)}_{n}}{D^{(r)}_{n}}\frac{z^{2n+1}}{2}\left[P_{n+1}(z^{-1};r+2)P_{n}(-z^{-1};r+2)-P_{n+1}(-z^{-1};r+2)P_{n}(z^{-1};r+2)\right],$
(5.49)
$R_{n}(z^{2};s)=\frac{E^{(s-2)}_{n}}{E^{(s)}_{n}}\frac{z^{2n+1}}{2}\left[S_{n+1}(z^{-1};s-2)S_{n}(-z^{-1};s-2)-S_{n+1}(-z^{-1};s-2)S_{n}(z^{-1};s-2)\right].$

###### Proof.

We only prove (5.48) as (5.49) can be proven in a similar way. From (5.14) we
have

(5.50)
$Q_{n}(z^{2};r)=\frac{1}{D^{(r)}_{n}}\mathcal{D}_{n}[w(\zeta)\zeta^{-r}(z^{2}-\zeta^{-2})].$

Using $z-\zeta^{-1}=-\zeta^{-1}z(z^{-1}-\zeta)$,
$z+\zeta^{-1}=-\zeta^{-1}z(-z^{-1}-\zeta)$, (5.50) and (2.12) we have

(5.51) $\begin{split}&D^{(r)}_{n}Q_{n}(z^{2};r)\\\
&=\frac{z^{2n}}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r-2}_{j}\prod_{j=1}^{n}(z^{-1}-\zeta_{j})(-z^{-1}-\zeta_{j})\prod_{1\leq
j<k\leq n}(\zeta_{k}-\zeta_{j})(\zeta^{-2}_{k}-\zeta^{-2}_{j}).\end{split}$

Now, we temporarily denote $z^{-1}$ and $-z^{-1}$ respectively by
$\zeta_{n+1}$ and $\zeta_{n+2}$, and thus

(5.52) $\begin{split}&D^{(r)}_{n}Q_{n}(z^{2};r)\\\
&=\frac{z^{2n}}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r-2}_{j}\frac{1}{\zeta_{n+2}-\zeta_{n+1}}\prod_{1\leq
j<k\leq n+2}(\zeta_{k}-\zeta_{j})\prod_{1\leq j<k\leq
n}(\zeta^{-2}_{k}-\zeta^{-2}_{j})\\\
&=-\frac{1}{2}\frac{z^{2n+1}}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r-2}_{j}\prod_{1\leq
j<k\leq n+2}(\zeta_{k}-\zeta_{j})\prod_{1\leq j<k\leq
n}(\zeta^{-2}_{k}-\zeta^{-2}_{j})\\\
&=-\frac{1}{2}\frac{z^{2n+1}}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r-2}_{j}\underset{\begin{subarray}{c}1\leq
j\leq n\\\ 1\leq k\leq
n+2\end{subarray}}{\det}\begin{pmatrix}P_{k-1}(\zeta_{j};r+2)\\\
P_{k-1}(z^{-1};r+2)\\\
P_{k-1}(-z^{-1};r+2)\end{pmatrix}\underset{\begin{subarray}{c}1\leq j\leq n\\\
1\leq k\leq
n\end{subarray}}{\det}\\{Q_{k-1}(\zeta^{-2}_{j};r+2)\\}.\end{split}$

We have used the offsets to be $r+2$, due to presence of $\zeta^{-r-2}_{j}$
and the fact that we are about to use the bi-orthogonality property of $P$’s
and $Q$’s. We have

(5.53) $\begin{split}&D^{(r)}_{n}Q_{n}(z^{2};r)\\\
&=-\frac{1}{2}z^{2n+1}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r-2}_{j}\underset{\begin{subarray}{c}1\leq
j\leq n\\\ 1\leq k\leq
n+2\end{subarray}}{\det}\begin{pmatrix}P_{k-1}(\zeta_{j};r+2)\\\
P_{k-1}(z^{-1};r+2)\\\
P_{k-1}(-z^{-1};r+2)\end{pmatrix}\prod_{j=1}^{n}Q_{j-1}(\zeta^{-2}_{j};r+2)\\\
&=-\frac{1}{2}z^{2n+1}\underset{\begin{subarray}{c}1\leq j\leq n\\\ 1\leq
k\leq
n+2\end{subarray}}{\det}\begin{pmatrix}\displaystyle\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta}{2\pi\textrm{i}\zeta}w(\zeta)\zeta^{-r-2}P_{k-1}(\zeta;r+2)Q_{j-1}(\zeta^{-2};r+2)\\\
P_{k-1}(z^{-1};r+2)\\\ P_{k-1}(-z^{-1};r+2)\end{pmatrix}\\\
&=-\frac{1}{2}z^{2n+1}\det\begin{pmatrix}h^{(r+2)}_{0}&\cdots&0&0&0\\\
\vdots&\ddots&\vdots&\vdots&\vdots\\\ 0&\cdots&h^{(r+2)}_{n-1}&0&0\\\
P_{0}(z^{-1};r+2)&\cdots&P_{n-1}(z^{-1};r+2)&P_{n}(z^{-1};r+2)&P_{n+1}(z^{-1};r+2)\\\
P_{0}(-z^{-1};r+2)&\cdots&P_{n-1}(-z^{-1};r+2)&P_{n}(-z^{-1};r+2)&P_{n+1}(-z^{-1};r+2)\\\
\end{pmatrix}.\end{split}$

Therefore

(5.54)
$D^{(r)}_{n}Q_{n}(z^{2};r)=-\frac{1}{2}z^{2n+1}\prod_{j=0}^{n-1}h^{(r+2)}_{j}\left[P_{n}(z^{-1};r+2)P_{n+1}(-z^{-1};r+2)-P_{n+1}(z^{-1};r+2)P_{n}(-z^{-1};r+2)\right].$

Noticing that $\prod_{j=0}^{n-1}h^{(r+2)}_{j}=D^{(r+2)}_{n}$ proves (5.48). ∎

######  Remark 5.10.

Since

$z^{n+1}P_{n+1}(z^{-1};\cdot)\Bigg{|}_{z=0}=z^{n}P_{n}(z^{-1};\cdot)\Bigg{|}_{z=0}=1,\quad\text{and}\quad-z^{n+1}P_{n+1}(-z^{-1};\cdot)\Bigg{|}_{z=0}=z^{n}P_{n}(-z^{-1};\cdot)\Bigg{|}_{z=0}=(-1)^{n},$

we observe that (5.48) is compatible with (4.135). A similar observation shows
the compatibility of (5.49) with (4.136).

###  5.3. The Christoffel-Darboux Identity

Having defined the reproducing kernel via a sum over products of bi-orthogonal
polynomials and deduced that these polynomials satisfy third order difference
equations we seek the explicit evaluation of the sum involved in the kernel.
The well known Christoffel-Darboux sum in the Toeplitz result can be found in
Prop. 2.5 of [35].

######  Theorem 5.11.

The Christoffel-Darboux identity for the $2j-k$ and $j-2k$ systems can be
respectively written as

(5.55)
$K_{n}(z^{2}_{2},z_{1};r)=\frac{1}{2}\frac{D_{n}^{(r+2)}}{D_{n+1}^{(r)}}\frac{z^{2n+1}_{2}}{z^{2}_{1}-z^{-2}_{2}}\det\begin{pmatrix}P_{n}(-z^{-1}_{2};r+2)&P_{n+1}(-z^{-1}_{2};r+2)&P_{n+2}(-z^{-1}_{2};r+2)\\\\[2.0pt]
P_{n}(z^{-1}_{2};r+2)&P_{n+1}(z^{-1}_{2};r+2)&P_{n+2}(z^{-1}_{2};r+2)\\\\[2.0pt]
P_{n}(z_{1};r+2)&P_{n+1}(z_{1};r+2)&P_{n+2}(z_{1};r+2)\\\ \end{pmatrix},$

and

(5.56)
$L_{n}(z_{2},z^{2}_{1};s)=\frac{1}{2}\frac{E_{n}^{(s-2)}}{E_{n+1}^{(s)}}\frac{z^{2n+1}_{1}}{z^{-2}_{1}-z^{2}_{2}}\det\begin{pmatrix}S_{n}(z^{-1}_{1};s-2)&S_{n+1}(z^{-1}_{1};s-2)&S_{n+2}(z^{-1}_{1};s-2)\\\\[2.0pt]
S_{n}(-z^{-1}_{1};s-2)&S_{n+1}(-z^{-1}_{1};s-2)&S_{n+2}(-z^{-1}_{1};s-2)\\\\[2.0pt]
S_{n}(z_{2};s-2)&S_{n+1}(z_{2};s-2)&S_{n+2}(z_{2};s-2)\\\ \end{pmatrix}.$

###### Proof.

Using (5.24) we have

(5.57)
$\begin{split}D^{(r)}_{n+1}K_{n}(z^{2}_{2},z_{1};r)&=\mathcal{D}_{n}[w(\zeta)\zeta^{-r}(z_{1}-\zeta)(z^{2}_{2}-\zeta^{-2})]\\\
&=\mathcal{D}_{n}[w(\zeta)\zeta^{-r}(z_{1}-\zeta)(z_{2}-\zeta^{-1})(z_{2}+\zeta^{-1})]\\\
&=\mathcal{D}_{n}[w(\zeta)\zeta^{-r}(z_{1}-\zeta)z^{2}_{2}\zeta^{-2}(z^{-1}_{2}-\zeta)(-z^{-1}_{2}-\zeta)]\\\
&=z^{2n}_{2}\mathcal{D}_{n}[w(\zeta)\zeta^{-r-2}(z_{1}-\zeta)(z^{-1}_{2}-\zeta)(-z^{-1}_{2}-\zeta)].\end{split}$

Let us temporarily denote $-z^{-1}_{2}$, $z^{-1}_{2}$, and $z_{1}$
respectively by $\zeta_{n+1}$, $\zeta_{n+2}$, and $\zeta_{n+3}$. Note that

(5.58) $\begin{split}\prod_{1\leq j<k\leq
n+3}(\zeta_{k}-\zeta_{j})=&(\zeta_{n+3}-\zeta_{n+2})(\zeta_{n+3}-\zeta_{n+1})(\zeta_{n+2}-\zeta_{n+1})\prod_{j=1}^{n}(\zeta_{n+3}-\zeta_{j})(\zeta_{n+2}-\zeta_{j})(\zeta_{n+1}-\zeta_{j})\prod_{1\leq
j<k\leq n}(\zeta_{k}-\zeta_{j}).\end{split}$

We have

$\begin{split}&D^{(r)}_{n+1}K_{n}(z^{2}_{2},z_{1};r)=z^{2n}_{2}\mathcal{D}_{n}[w(\zeta)\zeta^{-r-2}(z_{1}-\zeta)(z^{-1}_{2}-\zeta)(-z^{-1}_{2}-\zeta)]\\\
&=\frac{z_{2}^{2n}}{n!}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r-2}_{j}\prod_{j=1}^{n}(z_{1}-\zeta_{j})(z_{2}^{-1}-\zeta_{j})(-z_{2}^{-1}-\zeta_{j})\prod_{1\leq
j<k\leq n}(\zeta_{k}-\zeta_{j})(\zeta^{-2}_{k}-\zeta^{-2}_{j})\\\
&=\frac{z_{2}^{2n+1}}{n!2(z_{1}-z^{-1}_{2})(z_{1}+z^{-1}_{2})}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r-2}_{j}\prod_{1\leq
j<k\leq n+3}(\zeta_{k}-\zeta_{j})\prod_{1\leq j<k\leq
n}(\zeta^{-2}_{k}-\zeta^{-2}_{j})\\\
&=\frac{z_{2}^{2n+1}}{n!2(z_{1}-z^{-1}_{2})(z_{1}+z^{-1}_{2})}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r-2}_{j}\underset{\begin{subarray}{c}1\leq
j\leq n\\\ 1\leq k\leq
n+3\end{subarray}}{\det}\begin{pmatrix}P_{k-1}(\zeta_{j};r+2)\\\\[2.0pt]
P_{k-1}(-z_{2}^{-1};r+2)\\\\[2.0pt] P_{k-1}(z_{2}^{-1};r+2)\\\\[2.0pt]
P_{k-1}(z_{1};r+2)\end{pmatrix}\underset{\begin{subarray}{c}1\leq j\leq n\\\
1\leq k\leq n\end{subarray}}{\det}\\{Q_{k-1}(\zeta^{-2}_{j};r+2)\\}\\\
&=\frac{z_{2}^{2n+1}}{2(z^{2}_{1}-z^{-2}_{2})}\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{1}}{2\pi\textrm{i}\zeta_{1}}\cdots\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta_{n}}{2\pi\textrm{i}\zeta_{n}}\prod_{j=1}^{n}w(\zeta_{j})\zeta^{-r-2}_{j}\underset{\begin{subarray}{c}1\leq
j\leq n\\\ 1\leq k\leq
n+3\end{subarray}}{\det}\begin{pmatrix}P_{k-1}(\zeta_{j};r+2)\\\\[2.0pt]
P_{k-1}(-z_{2}^{-1};r+2)\\\\[2.0pt] P_{k-1}(z_{2}^{-1};r+2)\\\\[2.0pt]
P_{k-1}(z_{1};r+2)\end{pmatrix}\prod_{j=1}^{n}Q_{j-1}(\zeta^{-2}_{j};r+2)\\\
\end{split}$
$\begin{split}&=\frac{z_{2}^{2n+1}}{2(z^{2}_{1}-z^{-2}_{2})}\underset{\begin{subarray}{c}1\leq
j\leq n\\\ 1\leq k\leq
n+2\end{subarray}}{\det}\begin{pmatrix}\displaystyle\int_{{\mathbb{T}}}\frac{\textrm{d}\zeta}{2\pi\textrm{i}\zeta}w(\zeta)\zeta^{-r-2}P_{k-1}(\zeta;r+2)Q_{j-1}(\zeta^{-2};r+2)\\\\[2.0pt]
P_{k-1}(-z_{2}^{-1};r+2)\\\\[2.0pt] P_{k-1}(z_{2}^{-1};r+2)\\\\[2.0pt]
P_{k-1}(z_{1};r+2)\end{pmatrix}\\\
&=\frac{z_{2}^{2n+1}}{2(z^{2}_{1}-z^{-2}_{2})}\det\begin{pmatrix}h^{(r+2)}_{0}&\cdots&0&0&0&0\\\
\vdots&\ddots&\vdots&\vdots&\vdots&\vdots\\\
0&\cdots&h^{(r+2)}_{n-1}&0&0&0\\\\[2.0pt]
P_{0}(-z_{2}^{-1};r+2)&\cdots&P_{n-1}(-z_{2}^{-1};r+2)&P_{n}(-z_{2}^{-1};r+2)&P_{n+1}(-z_{2}^{-1};r+2)&P_{n+2}(-z_{2}^{-1};r+2)\\\\[2.0pt]
P_{0}(z_{2}^{-1};r+2)&\cdots&P_{n-1}(z_{2}^{-1};r+2)&P_{n}(z_{2}^{-1};r+2)&P_{n+1}(z_{2}^{-1};r+2)&P_{n+2}(z_{2}^{-1};r+2)\\\\[2.0pt]
P_{0}(z_{1};r+2)&\cdots&P_{n-1}(z_{1};r+2)&P_{n}(z_{1};r+2)&P_{n+1}(z_{1};r+2)&P_{n+2}(z_{1};r+2)\\\
\end{pmatrix}.\end{split}$

Now (5.55) follows immediately by noticing
$\prod_{j=0}^{n-1}h^{(r+2)}_{j}=D^{(r+2)}_{n}$. ∎

######  Remark 5.12.

We have found that the sum is most simply evaluated in terms of just the
polynomials $P_{n}$ and is therefore unsymmetrical with respect to $Q_{n}$. It
is possible to replace certain terms in this determinant by $Q_{n}$ using
(5.48) but not all in a symmetrical way.