Differential $t\overline{t}$ cross-section measurements using boosted top
quarks with 139 fb-1 of ATLAS data

Petr Jačka

on behalf of the ATLAS Collaboration

> Preliminary results of two differential cross-section measurements in the
> ATLAS experiment at CERN are presented. Measurements are focused on the top-
> antitop quark pair production in the lepton+jets and the all-hadronic
> channels in the kinematic region with high transverse momentum top quarks.
> Both measurements use the full Run 2 data of 13 TeV proton-proton collisions
> from the Large Hadron Collider collected by the ATLAS detector in 2015-2018,
> corresponding to an integrated luminosity of 139 fb-1. The measured spectra
> are compared with the Standard Model predictions. They are also used to set
> limits on the Wilson coefficients of an effective field theory extension of
> the Standard Model.

> PRESENTED AT
>
>
>
>
> $14^{\mathrm{th}}$ International Workshop on Top Quark Physics
> (videoconference), 13–17 September, 2021
>

Copyright 2022 CERN for the benefit of the ATLAS Collaboration. CC-BY-4.0
license.

##  1 Introduction

This note summarizes preliminary results of two measurements of top-antitop
($t\overline{t}$) quark pair production differential cross-sections in the
ATLAS experiment [1]. Measurements are focused on high transverse momentum
($p_{\rm T}$) top quarks in two channels: lepton+jets [2] and all-hadronic
[3]. Both measurements use the full Run 2 dataset of 13 TeV proton-proton
collisions from the Large Hadron Collider (LHC) collected by the ATLAS
detector in 2015-2018, corresponding to an integrated luminosity of 139 fb-1.
The $t\overline{t}$ production with high-$p_{\rm T}$ (boosted) top quarks is
sensitive to deviations from the Standard Model (SM) prediction. This
motivates for precise measurements in this topology. Results are compared with
SM predictions and are used to set limits on the Wilson coefficients of an
effective field theory (EFT) extension of the SM.

##  2 Reconstruction of top-antitop pairs

The two measurements use different strategies to reconstruct $t\overline{t}$
pairs. From now on, top quarks decaying into one $b$-quark, one charged lepton
and one neutrino are referred to as leptonic top quarks. Top quarks decaying
into one $b$-quark and two additional quarks are referred to as hadronic top
quarks.

The lepton+jets measurement is focused on events with one hadronic top and one
leptonic top quark. The hadronic top quark is reconstructed by large-$R$
($R=1.0$) anti-$k_{t}$ jets, reclustered from $R=0.4$ anti-$k_{t}$ particle-
flow jets. Large-$R$ jets are required to have $p_{\rm T}>355$ GeV,
$|\eta|<2.0$, and $|m-170~{}\mathrm{GeV}|<50$ GeV, where $\eta$ is
pseudorapidity and $m$ is the large-$R$ jet mass. In addition, the large-$R$
jet must contain at least one $R=0.4$ jet identified as coming from the
hadronization of $b$-quark ($b$-tagged). The leptonic top quark is
reconstructed from a charged lepton (electron or muon), a $b$-tagged $R=0.4$
jet, and a reconstructed neutrino.

The all-hadronic measurement is focused on events with two hadronic top
quarks. They are reconstructed using large-$R$ anti-$k_{t}$ jets, clustered
directly from calorimeter clusters. Jets are ordered according to $p_{\rm T}$.
The leading jet is required to have
$p_{\rm T}>500$ GeV and the subleading jet is required to have $p_{\rm T}>350$
GeV. Both jets are required to have $|\eta|<2.0$ and
$|m-172.5~{}\mathrm{GeV}|<50$ GeV. In addition, both jets are required to be
$b$-tagged and top-tagged.

The selection requirements above are used to define region of interest and to
reduce background contributions. The most relevant sources of the background
are listed in Table 1. Both measurements use data-driven techniques to
determine multijet background originating mainly from strong interactions of
the colliding protons. Contributions of other background sources are
determined by Monte Carlo simulations normalised to theoretical predictions of
their cross-sections.

Table 1: Main background processes ordered by their significance. Lepton+jets | All-hadronic
---|---
Single-top | Multijet
$t\overline{t}$ +$X(X=W,Z,H$) |  $t\overline{t}$ non-allhad
Multijet | Single-top
Others |  $t\overline{t}$ +$X(X=W,Z,H)$

Figure 1 shows the reconstructed top $p_{\rm T}$ distributions in both
channels. The spectra are steeply falling with $p_{\rm T}$ in both channels
and both measurements are reaching 2 TeV. The data distributions are overlayed
with the SM prediction, which is the sum of the signal and all background
contributions. Predictions are normalized to match data normalization.
Therefore, only the shapes of distributions can be compared.

Figure 1: Reconstructed hadronic top $p_{\rm T}$ in the lepton+jets channel
(left) [2] and leading top $p_{\rm T}$ in the all-hadronic channel (right)
[3]. Predictions are normalized to match data normalization. The prediction is
the sum of the signal and the background contributions.

##  3 Unfolded results

Reconstructed spectra are unfolded to a particle-level fiducial phase space
designed to keep the measured phase-space as close as possible to that
selected by the event selection. In addition, the all-hadronic analysis unfold
spectra to a parton-level fiducial phase defined by requiring that the leading
(subleading) top quark has
$p_{\rm T}>500(350)~{}\mathrm{GeV}$.

Figure 2: Particle-level fiducial phase-space differential cross-sections as a
function of hadronic top $p_{\rm T}$ in the lepton+jets channel (left) [2],
and the $p_{\rm T}$ of the second-leading top-quark jet in the all-hadronic
channel (right) [3]. The distribution of the the second-leading top-quark
$p_{\rm T}$ is normalized to unity. Data points are placed in the center of
each bin of measured spectra. The bands around data points indicate the total
and the statistical uncertainty in each bin. The data are compared to multiple
NLO+PS predictions represented by lines.

Both analyses use a Bayesian iterative method with 2-4 iterations to
regularize the unfolding problem. The all-hadronic measurements are further
normalized to unity to cancel out systematic uncertainties in the
normalization. Distributions at both levels are compared with $t\overline{t}$
next-to-leading order + parton-showering (NLO+PS) generators: POWHEG+Pythia8,
POWHEG+Herwig7, and MadGraph5 aMC@NLO+Pythia8 with various configurations. In
addition, the parton level spectra are compared with the SM fixed order next-
to-next-to leading order (NNLO) $t\overline{t}$ differential cross-section
prediction. Figure 2 shows the measured particle-level differential cross-
sections as a function of hadronic top $p_{\rm T}$ in the lepton+jets channel,
and the $p_{\rm T}$ of the subleading top-quark jet in the all-hadronic
channel. A small discrepancy is observed between data and predictions as well
as a mild inconsistency between NLO+PS generators.

Figure 3 (right) shows a comparison of the all-hadronic cross-section as a
function of leading top quark transverse momentum with the SM fixed-order
predictions and the POWHEG+Pythia8 predictions. All predictions are in good
agreement with data. However, the NNLO prediction tends to be closer to data
than other predictions. In addition, the NNLO prediction has smaller scale
uncertainties with respect to the NLO prediction. This demonstrates a better
precision and stability of higher-order predictions.

Figure 3: Two-dimensional limits on the EFT coefficients extending the
Standard Model (left)[2] and comparison of the NLO, NNLO, and POWHEG+Pythia8
predictions with measured parton-level fiducial phase-space absolute
differential cross-sections for the $p_{\rm T}$ of the leading top-quark
(right) [3].

Measurements of top-quark differential cross-sections are further used to
place constraints on Wilson coefficients $C_{i}$ in the EFT extension of the
SM model defined as

$\mathcal{L}_{\textrm{SM+EFT}}=\mathcal{L}_{\textrm{SM}}+C_{tG}\cdot\mathcal{O}_{tG}/\Lambda^{2}+C_{tq}^{(8)}\cdot\mathcal{O}_{tq}^{(8)}/\Lambda^{2},$

where the $\mathcal{O}_{tG}$ and $\mathcal{O}_{tq}^{(8)}$ operators induce
anomalous top-gluon and top-quark couplings and the energy scale $\Lambda$ is
set to 1 TeV (Figure 3 left). The measured values of the coefficients are
consistent with zero.

##  4 Conclusions

Preliminary results of two $t\overline{t}$ differential cross-sections
measurements in the ATLAS experiment at the LHC are presented. The
measurements are focused on high-$p_{\rm T}$ top quarks in the lepton+jets and
all-hadronic channels. Both measurements improved precision with respect to
the previous measurements. In general, good agreement is observed between
NLO+PS Monte Carlo simulations and the measured spectra. However, tensions are
observed in distributions sensitive to radiations. In addition, measurements
from the all-hadronic channel are compared with the fixed order NNLO
predictions. NNLO predictions tend to be closer to the measured spectra with
respect to NLO+PS predictions. Limits are set on selected coefficients of an
effective field theory extending the Standard Model. Coefficients are found to
be consistent with zero.

## References

  * [1] ATLAS Collaboration, 2008 JINST 3 S08003
  * [2] ATLAS Collaboration, ATLAS-CONF-2021-031 (2021), https://cds.cern.ch/record/2777237
  * [3] ATLAS Collaboration, ATLAS-CONF-2021-050 (2021), https://cds.cern.ch/record/2782534


*15cm1.6182 2.5cm** 3cm** [nearest] subsubsection subsubsection subsubsection bookstyle bookstyle0 bookstyle0 bookstyle chapter both shownumber . section rightshownumber . subsection rightshownumber . subsubsectionrightshownumber . toc bothContents lof bothList of Figures lot bothList of Tables bib both index bothIndex plain0 plain0

*


# Depolarization for quantum channels with higher symmetries

A B Klimov1 and L L Sánchez-Soto2 1 Departamento de Física, Universidad de
Guadalajara, 44420 Guadalajara, Jalisco, Mexico  2 Departamento de Óptica,
Facultad de Física, Universidad Complutense, 28040 Madrid, Spain

###### Abstract

The depolarization channel is usually modelled as a quantum operation that
destroys all input information, replacing it by a completely chaotic state.
For qubits this has a quite intuitive interpretation as a shrinking of the
Bloch sphere. We propose a way to deal with depolarizing dynamics (in the
Markov approximation) for systems with arbitrary symmetries.

###### pacs:

03.65.Yz,03.67.Hk,42.50.Lc

Quantum systems evolve via unitary operations determined by the Schrödinger
equation. This is also true for composite systems, e.g., a small quantum
system $\mathcal{S}$ that is surrounded by some environment $\mathcal{E}$,
with which it interacts. However, while the evolution of the total system is
described by a unitary operation $\hat{U}_{\mathcal{SE}}(t)$, the dynamics of
the system $\mathcal{S}$ alone (obtained by tracing out the uncontrollable
degrees of freedom of $\mathcal{E}$) is, in general, no longer unitary
[Breuer:2007, Weiss:2008, Alicki:2007]. The system-environment interaction
leads to entanglement between them, and this is reflected into the fact that
$\hat{U}_{\mathcal{SE}}(t)\neq\hat{U}_{\mathcal{S}}(t)\otimes\hat{U}_{\mathcal{E}}(t)$.
From the perspective of quantum information processing, such an interaction is
undesirable and causes errors and noise in the system [Nielsen:2000].

The dynamics of $\mathcal{S}$ can be described by a finite-time trace-
preserving completely positive map (CPM) [Stinespring:1955, Choi:1972] that
transforms input states $\hat{\varrho}_{\mathrm{in}}=\hat{\varrho}(0)$ into
output states $\hat{\varrho}_{\mathrm{out}}=\hat{\varrho}(t)$, i.e.,

$\varrho_{\mathrm{in}}\mapsto\hat{\varrho}_{\mathrm{out}}=\mathcal{E}_{t}(\hat{\varrho}_{\mathrm{in}})\,,$
(1)

which is also known as a quantum channel. One may think of the environment as
extracting information from the system, as it will typically map pure states
into mixed states.

This noise process can be also described by a quantum operation involving only
operators of the system of interest. This is called a Kraus decomposition
[Kraus:1983] and has the form (omitting all the unnecessary subscripts)

$\mathcal{E}(\hat{\varrho})=\sum_{r}\hat{K}_{r}\,\hat{\varrho}\,\hat{K}_{r}^{\dagger}\,,$
(2)

where the Kraus operators satisfy the condition

$\sum_{r}\hat{K}_{r}\,\hat{K}_{r}^{\dagger}=\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}}\,,$ (3)

which ensures that unit trace is preserved for all times.

A channel is Markovian when the coupling coupling of the system $\mathcal{S}$
with the environment $\mathcal{E}$ can be treated under the Markov and Born
approximations. The evolution of the state at a given instant is then fully
determined by the state at that instant, so the process ha no “memory” of its
past. This is a commonly used approximation in quantum optics and leads to the
well-known Lindblad form of a master equation [Lindblad:1976, Gorini:1976].
For these Markovian channels, one can always write

$\mathcal{E}_{t}(\hat{\varrho})=e^{\mathcal{L}t}\,\hat{\varrho}(0)\,,$ (4)

where the Lindblad superoperator $\mathcal{L}$ is

$\mathcal{L}(\hat{\varrho})=-i[\hat{H},\hat{\varrho}]+\frac{1}{2}\sum_{r}a_{r}\left([\hat{L}_{r},\hat{\varrho}\hat{L}_{r}^{\dagger}]+[\hat{L}_{r}\hat{\varrho},\hat{L}_{r}^{\dagger}]\right)\,.$
(5)

Here $\hat{H}$ is the Hamiltonian of the undamped system $\mathcal{S}$,
$\hat{L}_{r}$ are system operators defined to model the effective dissipative
interaction with the environment, and $a_{r}\geq 0$ are constants that account
for decoherence rates. In this form, $\mathcal{L}$ appears as the generator of
a CMP. In fact, expanding equation (5) to first order in the short-time
interval $\tau$ one can immediately find the corresponding Kraus operators
[Shabani:2005]

$\displaystyle\hat{K}_{0}$ $\displaystyle=$
$\displaystyle\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}-\tau\left(i\hat{H}+\frac{1}{2}\sum_{r}\hat{L}_{r}^{\dagger}\,\hat{L}_{r}\right)\,$
$\displaystyle\hat{K}_{r}$ $\displaystyle=$
$\displaystyle\sqrt{\tau}\,\hat{L}_{r}\,.$

For obvious reasons, the characterization and classification of these maps
have attracted a lot of interest in recent years [Keyl:2002]. The majority of
the results obtained so far relate to two specific classes: the qubit channels
and the bosonic Gaussian channels [Caruso:2007]. In this paper, we shall be
mainly concerned with the former.

In classical computation, the only error that can occur is the bit flip
$0\leftrightarrow 1$. In quantum computation, however, the existence of
superposition states brings also the possibility of other basic errors for a
single qubit. They are the phase flip and the bit-phase flip. The first
changes the phase of the state, and the latter combines phase and bit flips.
The set of Kraus operators for each one of these channels is given by

$\hat{K}_{0}=\sqrt{1-p/2}\,\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}}\,,\qquad\hat{K}^{j}_{1}=\sqrt{p/2}\,\hat{\sigma}_{j}\,,$
(7)

where $j=x$ gives the bit flip, $j=z$ the phase flip and $j=y$ the phase-bit
flip. They can also interpreted as corresponding to a probability $1-p/2$ of
remaining in the same state, and a probability $p/2$ of having an error. It is
not difficult to represent these channels in terms of a master equation of the
form (5): the associated Lindblad superoperator $\hat{L}^{j}_{1}$ turns out to
be the corresponding Pauli matrix $\hat{\sigma}_{j}$.

The environment-induced noise is modelled using various simplified approaches.
Here, we concentrate in the depolarization channel, which employs unbiased
noise generating bit flip errors and phase flip errors and is represented by
the CPM [Bennett:1997]

$\mathcal{E}(\hat{\varrho})=(1-\hat{\varrho})+\frac{p}{3}(\hat{\sigma}_{x}\,\hat{\varrho}\,\hat{\sigma}_{x}+\hat{\sigma}_{y}\,\hat{\varrho}\,\hat{\sigma}_{y}+\hat{\sigma}_{z}\,\hat{\varrho}\,\hat{\sigma}_{z})\,.$
(8)

Note, however, that, since the three interaction channels corresponding to bit
error, flip error, and phase error do not commute, one could rightly argue
that the incoherent addition of these channels in the CMP (8) is, at least,
questionable.

The Kraus decomposition of this channel reads as

$\displaystyle\hat{K}_{0}=\sqrt{1-p}\,\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}},$ (9)
$\displaystyle\displaystyle\hat{K}_{1}=\frac{p}{3}\,\hat{\sigma}_{x},\qquad\hat{K}_{2}=\frac{p}{3}\,\hat{\sigma}_{y},\qquad\hat{K}_{3}=\frac{p}{3}\,\hat{\sigma}_{z}\,,$

while the associated Lindblad equation is

$\dot{\hat{\varrho}}=-\Gamma\left(\hat{\varrho}-\frac{1}{2}\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}}\right)\,,$ (10)

where we have omitted the free evolution of the system, since it is irrelevant
for our purposes here, and $\Gamma$ is a constant. If we use the standard
Bloch parametrization for the density matrix

$\hat{\varrho}=\frac{1}{2}\left(\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}}+\bm{s}\cdot\hat{\bm{\sigma}}\right)\,,$ (11)

we immediately get

$\dot{\bm{s}}=-\Gamma\bm{s}\,,$ (12)

which clearly shows that the action of the channel is to contract the sphere
with a lifetime $\Gamma^{-1}$. This is precisely the idea behind
depolarization: for long times the system will end in a fully depolarized or
chaotic state, whose density matrix is diagonal
$\hat{\varrho}_{\mathrm{unpol}}=\frac{1}{2}\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}}$.

For $n$ qubits, a possible extension is to assume that the error operators can
be represented by the $n$-qubit Pauli group [Nielsen:2000]

$\mathcal{P}_{n}=\\{\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}},\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z}\\}^{\otimes
n}\,,$ (13)

where $\otimes n$ denotes the $n$-fold tensor product. This means that each
qubit is acted by the identical independent depolarizing channels. However,
alternative models, such as collective or correlated depolarization, have been
proposed [Reina:2002, Banaszek:2004, Ball:2004, Ball:2005, Ban:2006], which do
not fit in this simple extension. The point we wan to stress is that the final
result depends obviously of the symmetry of the channel and this, in turn,
depends on physical considerations.

The purpose of this work is to address this problem and identify the proper
depolarizing channel for more general situations. To this end, consider a
system whose Hamiltonian is a function of a symmetry Lie algebra
$\mathfrak{A}$. In order to maintain the discussion as simple as possible, we
assume that this algebra is semisimple and denote by $\Delta$ the set of
nonzero roots [Erdmann:2006dg]. We use the standard Cartan-Weyl basis
$\\{\hat{h}_{i},\hat{e}_{\alpha}\\}$ in terms of which we have the commutation
relations

$[\hat{h}_{i},\hat{h}_{j}]=0\,,\qquad[\hat{h}_{i},\hat{e}_{\alpha}]=\alpha(\hat{h}_{i})\,\hat{e}_{\alpha}\,,\qquad[\hat{e}_{\alpha},\hat{e}_{\beta}]=N_{\alpha\beta}\,\hat{e}_{\alpha+\beta}\,,$
(14)

where the last one is valid only when $\alpha+\beta\in\Delta$. The operators
$\\{\hat{h}_{i}\\}$ ($i$ runs from 1 to $\ell$, where $\ell$ is the rank of
the group) constitute the Cartan subalgebra and may be taken diagonal in any
irreducible representation. On the other hand,
$\\{\hat{e}_{\alpha},\hat{e}_{-\alpha}\\}$ are raising and lowering operators
and we can always choose $\hat{e}_{\alpha}^{\dagger}=\hat{e}_{-\alpha}$.

The Hilbert space decomposes also into finite-dimensional subspaces

$\mathcal{H}=\bigoplus_{\lambda}\mathcal{H}_{\lambda}$ (15)

so that in each $\mathcal{H}_{\lambda}$ the operators of $\mathfrak{A}$ act
irreducibly. Let $|\mathbf{h};\lambda\rangle$ be an orthonormal basis in
$\mathcal{H}_{\lambda}$, where
$\mathbf{h}=(h_{1},\ldots,h_{j},\ldots,h_{\ell})$. Then, we have that

$\hat{h}_{j}|\mathbf{h};\lambda\rangle=h_{j}|\mathbf{h};\lambda\rangle\,.$
(16)

Notice also that the density operator of any quantum unpolarized state can be
written in terms of these invariant subspaces

$\hat{\varrho}_{\mathrm{unpol}}=\bigoplus_{\lambda}r_{\lambda}\,\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}}_{\lambda}\,,$ (17)

where $\hat{\leavevmode\hbox{\small 1\normalsize\kern-3.30002pt1}}_{\lambda}$
denotes the unity in the corresponding subspace and all the coefficients
$r_{\lambda}$ are real and nonnegative and they fulfill the unit-trace
condition.

Since all the error operators are in the algebra $\mathfrak{A}$, the only
possible Kraus operators are also elements of $\mathfrak{A}$, and they induce
a local Lindblad equation fully analogous to (5). Very different processes can
be described in terms of this master equation. The first one is what can be
called a pure dephasing channel, represented by

$\hat{L}_{r}=\hat{h}_{r}\,.$ (18)

Such a map obviously preserves the diagonal operators:
$\mathcal{L}(\hat{h}_{r})=0$, and the asymptotic limit of the channel is given
by

$\lim_{n\rightarrow\infty}\mathcal{E}^{n}(\hat{\varrho})=\sum_{\mathbf{h},\lambda}\varrho_{\mathbf{h}\mathbf{h}}(\lambda)\,|\mathbf{h};\lambda\rangle\langle\mathbf{h};\lambda|\,,$
(19)

where $\varrho_{\mathbf{h}\mathbf{h}}(\lambda)$ are just precisely the
occupation probabilities in each invariant subspace. For the case a symmetry
algebra su(2), generated by $\\{\hat{S}_{x},\hat{S}_{y},\hat{S}_{z}\\}$, the
archetypal example of this situation is

$\mathcal{L}(\hat{\varrho})=\hat{S}_{z}\,\hat{\varrho}\,\hat{S}_{z}-\frac{1}{2}(\hat{S}_{z}^{2}\,\hat{\varrho}+\hat{\varrho}\,\hat{S}_{z}^{2})=\hat{S}_{z}\,\hat{\varrho}\,\hat{S}_{z}-\hat{\varrho}\,,$
(20)

which is a standard way of accounting for process that lead to a loss of
coherence without changing the level populations [Briegel:1993xy].

Next we consider a generalized amplitude-damping channel, for which

$\hat{L}_{r}=\hat{e}_{-\alpha_{r}}\,.$ (21)

In the asymptotic limit, this leads to the ground state in each invariant
subspace

$\lim_{n\rightarrow\infty}\mathcal{E}^{n}(\hat{\varrho})=\sum_{\lambda}|\mathbf{h}_{\mathrm{min}};\lambda\rangle\langle\mathbf{h}_{\mathrm{min}};\lambda|\,,$
(22)

where $|\mathbf{h}_{\mathrm{min}};\lambda\rangle$ is the lowest-weight state
that is annihilated by all the lowering operators:
$\hat{e}_{-\alpha}|\mathbf{h}_{\mathrm{min}};\lambda\rangle=0$. It is clear
that now no Lindblad (neither Kraus) map preserves the diagonal operators. In
particular, for su(2) symmetry, the textbook example is

$\mathcal{L}(\hat{\varrho})=\hat{S}_{-}\hat{\varrho}\hat{S}_{+}-\frac{1}{2}(\hat{S}_{+}\hat{S}_{-}\hat{\varrho}+\hat{\varrho}\hat{S}_{+}\hat{S}_{-})\,.$
(23)

which is a standard model for damping [Agarwal:1974rm].

While these generalizations are more or less obvious, the corresponding one
for a depolarization channel is far from trivial. We claim that such a
situation must be described by the action of a Lindblad operator proportional
to $\hat{e}_{-\alpha_{r}}$ followed by other proportional to
$\hat{e}_{+\alpha_{r}}$. One can check that only in this way we asymptotically
get an unpolarized state

$\lim_{n\rightarrow\infty}\mathcal{E}^{n}(\hat{\varrho})=\sum_{\lambda}\Tr[\hat{\varrho}(\lambda)]\,\hat{\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}}_{\lambda}=\hat{\varrho}_{\mathrm{unpol}},$ (24)

where the trace operation is taken in each invariant subspace. The associated
Lindblad operator is

$\mathcal{L}(\hat{\varrho})=\frac{1}{2}\sum_{r}a_{r}\left(2\hat{e}_{-\alpha_{r}}\,\hat{\varrho}\,\hat{e}_{\alpha_{r}}+2\hat{e}_{\alpha_{r}}\,\hat{\varrho}\hat{e}_{-\alpha_{r}}-\\{\hat{e}_{\alpha_{r}},\hat{e}_{-\alpha_{r}}\\}\,\hat{\varrho}-\hat{\varrho}\,\\{\hat{e}_{\alpha_{r}},\hat{e}_{-\alpha_{r}}\\}\right)\,,$
(25)

and one can check that $\mathcal{L}(\hat{\varrho}_{\mathrm{unpol}})=0$. An
example of this situation for qubit systems parallels completely (25), but
with the corresponding spin-like operators is

$\mathcal{L}(\hat{\varrho})=\hat{S}_{-}\hat{\varrho}\hat{S}_{+}+\hat{S}_{+}\hat{\varrho}\hat{S}_{-}-\frac{1}{2}\left(\\{\hat{S}_{+},\hat{S}_{-}\\}\hat{\varrho}+\hat{\varrho}\\{\hat{S}_{+},\hat{e}_{-}\\}\right)\,.$
(26)

It is worth noting that such a Linblad operator appears as a limit case of
decaying into a bath at infinite temperature.

An effective depolarization channel is actually a common situation in driven
dissipative systems. Consider a typical evolution equation

$\dot{\hat{\varrho}}=-ig[\hat{S}_{x},\hat{\varrho}]+\gamma[\hat{S}_{-}\hat{\varrho}\hat{S}_{+}-\frac{1}{2}\left(\\{\hat{S}_{+},\hat{S}_{-}\\}\,\hat{\varrho}+\hat{\varrho}\,\\{\hat{S}_{+},S_{-}\\}\right)\,,$
(27)

which describes the decay (with rate $\gamma$) of an externally-driven (with
coupling constant $\kappa$) collective spin into a zero-temperature bath
[Agarwal:1974rm]. In the strong-pumping limit ($g\gg\gamma$), the above
equation can be easily diagonalized [Klimov:2000la]: it is enough to apply the
rotation $\hat{U}=\exp(i\pi\hat{S}_{y}/2)$ and to make the rotating wave
approximation. The final result is

$\displaystyle\dot{\hat{\varrho}}_{d}$ $\displaystyle=$ $\displaystyle-
ig[\hat{S}_{z},\hat{\varrho}_{d}]+\frac{\gamma}{2}\left(2\hat{S}_{z}\hat{\varrho}\hat{S}_{z}-\hat{S}_{z}^{2}\hat{\varrho}-\hat{\varrho}\hat{S}_{z}^{2}\right)$
(28) $\displaystyle+$
$\displaystyle\frac{\gamma}{2}\left(2\hat{S}_{-}\hat{\varrho}\hat{S}_{+}+2\hat{S}_{+}\hat{\varrho}\hat{S}_{-}-\\{\hat{S}_{+},\hat{S}_{-}\\}\,\hat{\varrho}-\hat{\varrho}\,\\{\hat{S}_{+},\hat{S}_{-}\\}\right)\,,$

where $\hat{\varrho}_{d}=\hat{U}\,\hat{\varrho}\,\hat{U}^{\dagger}$ is the
density matrix in the rotated frame. We can clearly observe the emergence a
pure dephasing (first line) and a depolarizing channel (second line).

In summary, what we expect to have accomplished in this paper is to provide a
construction of the depolarizing channel for systems with arbitrary
symmetries. This may be more than an academic curiosity for more involved
systems currently under investigation as candidates for quantum information
processing.

This work was supported by the Grant No 45704 of Consejo Nacional de Ciencia y
Tecnología (CONACyT) and the Spanish Research Directorate (Grant
FIS2005-06714). A. B. K. was also supported by the Spanish Sabbatical Program
(Grant SAB2006-0064).

## References

  * [1] Agarwal1974Agarwal:1974rm Agarwal G 1974 Quantum Statistical Theories of Spontaneous Emission and their Relation to other Approaches Vol. 70 of Springer Tracts in Modern Physics Springer.
http://dx.doi.org/10.1007/BFb0042382

  * [2] Alicki Lendi2007Alicki:2007 Alicki R Lendi K 2007 Quantum Dynamical Semigroups and Applications Vol. 286 of Lecture Notes in Physics Springer.
  * [3] Ball et al.2004Ball:2004 Ball J, Dragan A Banaszek K 2004 Phys. Rev. A 69, 042324.
  * [4] Ball Banaszek2005Ball:2005 Ball J L Banaszek K 2005 Open Sys. Inf. Dyn. 12, 121–131.
  * [5] Ban Shibata2006Ban:2006 Ban M Shibata F 2006 Phys. Lett. A 354, 35–39.
  * [6] Banaszek et al.2004Banaszek:2004 Banaszek K, Dragan A, Wasilewski W Radzewicz C 2004 Phys. Rev. Lett. 92, 257901.
  * [7] Bennett et al.1997Bennett:1997 Bennett C H, DiVincenzo D P Smolin J A 1997 Phys. Rev. Lett. 78, 3217–3220.
  * [8] Breuer Petruccione2007Breuer:2007 Breuer H P Petruccione F 2007 The Theory of Open Quantum Systems Oxford University Press Oxford.
  * [9] Briegel Englert1993Briegel:1993xy Briegel H J Englert B G 1993 Phys. Rev. A 47, 3311–3329.
  * [10] Caruso Giovannetti2007Caruso:2007 Caruso F Giovannetti V 2007 Phys. Rev. A 76, 042331.
  * [11] Choi1972Choi:1972 Choi M D 1972 Can. J. Math. 24, 520—529.
  * [12] Erdmann Wildon2006Erdmann:2006dg Erdmann K Wildon M 2006 Introduction to Lie Algebras Springer Berlin.
  * [13] Gorini et al.1976Gorini:1976 Gorini V, Kossakowski A Sudarshan E C G 1976 J. Math. Phys. 17, 821–825.
  * [14] Keyl2002Keyl:2002 Keyl M 2002 Phys. Rep. 369, 431–548.
  * [15] Klimov Sánchez-Soto2000Klimov:2000la Klimov A B Sánchez-Soto L L 2000 Phys. Rev. A 61, 063802.
  * [16] Kraus1983Kraus:1983 Kraus K 1983 States, Effects and Operations: Fundamental No- tions of Quantum Theorytions of Quantum Theory Springer Berlin.
  * [17] Lindblad1976Lindblad:1976 Lindblad G 1976 Commun. Math. Phys. 48, 119–130.
  * [18] Nielsen Chuang2000Nielsen:2000 Nielsen M A Chuang I L 2000 Quantum Computation and Quantum Information Cambridge University Press Cambridge.
  * [19] Reina et al.2002Reina:2002 Reina J H, Quiroga L Johnson N F 2002 Phys. Rev. A 65, 032326.
  * [20] Shabani Lidar2005Shabani:2005 Shabani A Lidar D A 2005 Phys. Rev. A 71, 020101.
  * [21] Stinespring1955Stinespring:1955 Stinespring W F 1955 Proc. Am. Math. Soc. 6, 211–216.
  * [22] Weiss2008Weiss:2008 Weiss U 2008 Quantum Dissipative Systems third edn World Scientific Singapore.
  * [23]


# Hankel determinants of the Cantor sequence

Zhi-Xiong WEN  School of Mathematics and Statistics
Huazhong University of Science and Technology
430074, Wuhan, P. R. China<EMAIL_ADDRESS>and  Wen WU∗ Department
of Mathematics
Hubei University
430062, Wuhan, P. R. China<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In the paper, we give the recurrent equations of the Hankel determinants of
the Cantor sequence, and show that the Hankel determinants as a double
sequence is $3$-automatic. With the help of the Hankel determinants, we prove
that the irrationality exponent of the Cantor number, i.e. the transcendental
number with Cantor sequence as its $b$-ary expansion, equals $2$.

###### Key words and phrases:

Cantor sequence, Hankel determinant, Automatic sequence, Irrationality
exponent, Padé approximation

###### 2000 Mathematics Subject Classification:

11J04, 11J82, 41A21

This work is supported by NSFC (Grant Nos. 11371156).
$*$ Wen Wu is the corresponding author.

##  1\. Introduction.

Let $\sigma$ be the _Cantor substitution_ defined over the alphabet
$\mathcal{A}=\\{a,b\\}$ by

$\sigma:a\mapsto aba,b\mapsto bbb.$

Since the word $\sigma^{n}(a)$ is a prefix of $\sigma^{n+1}(a)$, i.e.,
$\sigma^{n+1}(a)=\sigma^{n}(a)w$ where $w$ is a finite word over the alphabet
$\mathcal{A}$, the sequence of words $(\sigma^{n}(a))_{n\in\mathbb{N}}$
converges to the infinite sequence

$\mathbf{c}:=c_{0}c_{1}c_{2}\cdots\in\mathcal{A}^{\mathbb{N}},$

called the _Cantor sequence_. In this paper, we take $a=1$ and $b=0$. Then the
sequence $(c_{n})_{n\geq 0}$ takes the value 1 if the $3$-ary expansion of $n$
contains no ’1’, and 0 otherwise(see [2], [6]). Here are the first few terms:

$n$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | $\cdots$
---|---|---|---|---|---|---|---|---|---|---
$c_{n}$ | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | $\cdots$

The Cantor sequence is an _automatic sequence_ (see [2]), i.e., it can be
generated by a finite automaton. In detail, the Cantor sequence can be
recognized by the $3$-automaton in Figure 1 in direct reading with the initial
state $a$ and the output map $a\mapsto 1,b\mapsto 0$.

$a/1$start$b/0$0,210,1,2 Figure 1. Automaton Generating the Cantor Sequence.

The Cantor sequence can be recognized as the discretization of the Cantor
ternary set. In fact, we can construct the Cantor ternary set from the Cantor
sequence as a recurrent set, following the way introduced by F. M. Dekking in
[5].

In this paper, we discuss some property of the Hankel determinants of the
Cantor sequence.

###  1.1. Hankel determinants of Cantor sequence.

For a sequence of complex numbers $\mathbf{u}=u_{0}u_{1}\cdots u_{n}\cdots,$
the corresponding $(p,n)$-order _Hankel matrix_ $H_{n}^{p}$ is given by

$H_{n}^{p}=\begin{pmatrix}u_{p}&u_{p+1}&\cdots&u_{p+n-1}\\\
u_{p+1}&u_{p+2}&\cdots&u_{p+n}\\\ \cdots&\cdots&\cdots&\cdots\\\
u_{p+n-1}&u_{p+n}&\cdots&u_{p+2n-2}\end{pmatrix}$

where $n\geq 1$ and $p\geq 0$. And $|H_{n}^{p}|$ denote the determinant of the
matrix $H_{n}^{p}$.

Positive definiteness of Hankel matrices associated with a sequence are strong
connected the moment problem(see [11]). The properties of Hankel determinants
are also connnected to the combinatoric properties of the sequences and to the
Padé approximation(see [3], [4]). In [8], Kamae, Tamura and Wen studied the
properties of Hankel determinants for the Fibonacci word and give a
quantitative relation between the Hankel determinant and the Padé pair. Later,
Tamura [12] generalized the results for a class of special sequences.
Allouche, Peyrière, Wen and Wen studied the properties of Hankel determinants
$|\mathcal{E}_{n}^{p}|$ of the Thue-Morse sequence in [1]. They proved that
the Hankel determinants $|\mathcal{E}_{n}^{p}|$(modulo 2) recognized as a two-
dimensional sequence (or _double sequence_) was 2-automatic. Recently, Gou,
Wen and Wu [7] proved that the Hankel determinants (modulo 2) of regular
paper-folding sequence are also periodic.

Let $\Gamma_{n}^{p}$ be the $(p,n)$-order Hankel matrix of Cantor sequence.
Our purpose is to discuss the property of the two-dimensional sequence
$\\{|\Gamma_{n}^{p}|\\}_{n,p\geq 0}$. We have the following results.

Main Results. __

  1. (1)

_For each_ $p\geq 0$, the sequence $\\{|\Gamma_{n}^{p}|\\}_{n\geq 0}$ is
periodic(see Theorem 2).

  2. (2)

_The two-dimensional sequence(modulo_ $3$) $\\{|\Gamma_{n}^{p}|\\}_{n,p\geq
0}$ is $3$-automatic (see Theorem 3).

Figure 2 shows $|\Gamma_{n}^{p}|$ modulo $3$ for $1\leq n\leq 96$ and $0\leq
p\leq 127$, where $0$’s are replaced by blue squares, $1$’s are replaced by
green squares and $2$’s are replaced by red squares. Columns refer sequences
$\\{|\Gamma_{n}^{p}|\\}_{n\geq 0}$.

Figure 2. $|\Gamma^{p}_{n}|(\text{mod }3)$ for $1\leq n\leq 96$ and $0\leq
p\leq 127$.

This article is organized as follows. Definitions and preliminaries are given
in Section 2. In Section 3, we give recurrence formulae of Hankel determinants
(modulo $3$) of the Cantor sequence. Section 4 is mainly devoted to
characterize the periodicity and automaticity properties of Hankel
determinants (modulo $3$) of the Cantor sequence. In Section 5, we will prove
that the Padé approximants to the generating series of the Cantor sequence
exist and the irrationality exponent of the Cantor number and the difference
sequence are both equal to $2$.

##  2\. Preliminaries.

Assume that $M$ is an $n\times n$-matrix. Denote by $M^{t}$ the transpose of
$M$. Let $M^{(i)}$ be the $n\times(n-1)$-matrix obtained by deleting the
$i$-th _column_ of $M$, and $M_{(i)}$ be the $(n-1)\times n$ -matrix obtained
by deleting the $i$-th _row_ of $M$. $|M|$ denote the determinant of the
matrix $M$ and $\mathbf{0}_{m\times n}$ denote the $m\times n$ zero matrix.

It is easy to check that the Cantor sequence $\mathbf{c}=c_{0}c_{1}\cdots
c_{n}\cdots\in\\{0,1\\}^{\mathbb{N}}$ can also be defined by the following
recurrence equations:

(2.1) $c_{0}=1,~{}c_{3n}=c_{n},~{}c_{3n+1}=0,~{}c_{3n+2}=c_{n},\text{ for
}n\geq 0.$

For further consideration, we need another sequence $\mathbf{d}$ defined by
$d_{n}=c_{n}+c_{n+2}$, and simultaneously we have

(2.2) $d_{3n}=2c_{n},~{}d_{3n+1}=c_{n+1},~{}d_{3n+2}=c_{n}.$

The Hankel matrices of this difference sequences $\mathbf{d}$ is denoted by
$\Delta_{n}^{p}$. An easy observation show that

$\Delta_{n}^{p}=\Gamma_{n}^{p}+\Gamma_{n}^{p+2}.$

Let $K_{n}^{p}:=(u_{p+3(i+j-2)})_{1\leq i,j\leq n}$. When
$\mathbf{u}=\mathbf{c}$ is the Cantor sequence, by (2.1), we have for all
$n\geq 1,p\geq 0,$

(2.3) $K_{n}^{3p}=\Gamma_{n}^{p},~{}K_{n}^{3p+1}=\mathbf{0}_{n\times
n},~{}K_{n}^{3p+2}=\Gamma_{n}^{p}.$

When $\mathbf{u}=\mathbf{d}$, by (2.2), for all $n\geq 1,p\geq 0,$

(2.4)
$K_{n}^{3p}=2\Gamma_{n}^{p}\equiv-\Gamma_{n}^{p},~{}K_{n}^{3p+1}=\Gamma_{n}^{p+1},~{}K_{n}^{3p+2}=\Gamma_{n}^{p},$

where the symbol $\equiv$, unless otherwise stated, means equality modulo $3$
throughout this paper.

Now we will give an auxiliary lemma on matrices.

######  Lemma 1.

For all $n\geq 2,p\geq 0$,

$\left|\begin{matrix}\Gamma_{n}^{p}&\Gamma_{n}^{p+1}\\\
\Gamma_{n}^{p+1}&-\Gamma_{n}^{p}\end{matrix}\right|=(-1)^{n}|\Gamma_{n}^{p}|\cdot|\Delta_{n}^{p}|+(-1)^{n+1}|\Gamma_{n+1}^{p}|\cdot|\Delta_{n-1}^{p}|.$

###### Proof.

Suppose $\alpha_{p}^{n}$ is the column vector of the form
$(c_{p},c_{p+1},\cdots,c_{p+n-1})^{t}$, then

$\left|\begin{matrix}\Gamma_{n}^{p}&\Gamma_{n}^{p+1}\\\
\Gamma_{n}^{p+1}&-\Gamma_{n}^{p}\end{matrix}\right|=\left|\begin{matrix}\Gamma_{n}^{p}&\alpha_{p+n}^{n}&\alpha_{p}^{n}&\Gamma_{n}^{p+1}\\\
\mathbf{0}_{1\times n}&1&0&\mathbf{0}_{1\times n}\\\ \mathbf{0}_{1\times
n}&0&1&\mathbf{0}_{1\times n}\\\
\Gamma_{n}^{p+1}&\alpha_{p+n+1}^{n}&\mathbf{0}_{n\times
1}&-\Gamma_{n}^{p}\end{matrix}\right|.$

Note that
$(\Gamma_{n}^{p}~{}\alpha_{p+n}^{n})=(\alpha_{p}^{n}~{}\Gamma_{n}^{p+1})$, we
have

$\displaystyle\left|\begin{matrix}\Gamma_{n}^{p}&\Gamma_{n}^{p+1}\\\
\Gamma_{n}^{p+1}&-\Gamma_{n}^{p}\end{matrix}\right|$ $\displaystyle=$
$\displaystyle\left|\begin{matrix}\Gamma_{n}^{p}&\alpha_{p+n}^{n}&0&\mathbf{0}_{n\times
n}\\\ \mathbf{0}_{1\times n}&1&0&0\cdots 0~{}(-1)\\\ \mathbf{0}_{1\times
n}&0&1&0\cdots 0~{}~{}~{}~{}0~{}~{}\\\
\Gamma_{n}^{p+1}&\alpha_{p+n+1}^{n}&-\alpha_{p+1}^{n}&-\Delta_{n}^{p}\end{matrix}\right|$
$\displaystyle=$ $\displaystyle\left|\begin{matrix}1&0\\\
0&1\end{matrix}\right|\cdot\left|\begin{matrix}\Gamma_{n}^{p}&\mathbf{0}\\\
\Gamma_{n}^{p+1}&-\Delta_{n}^{p}\end{matrix}\right|$
$\displaystyle+(-1)^{n+1}\left|\begin{matrix}0&-1\\\
1&0\end{matrix}\right|\left|\begin{matrix}(\Gamma_{n+1}^{p})_{(n+1)}&\mathbf{0}_{n\times(n-1)}\\\
(\Gamma_{n+1}^{p+1})_{(n+1)}&-(\Delta_{n}^{p})^{(n)}\end{matrix}\right|.$

Since

$\displaystyle\left|\begin{matrix}(\Gamma_{n+1}^{p})_{(n+1)}&\mathbf{0}_{n\times(n-1)}\\\
(\Gamma_{n+1}^{p+1})_{(n+1)}&-(\Delta_{n}^{p})^{(n)}\end{matrix}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{matrix}(\Gamma_{n+1}^{p})_{(n+1)}&\mathbf{0}_{n\times(n-1)}\\\
\begin{array}[]{c}\mathbf{0}_{(n-1)\times n}\\\
(\alpha_{p+n+1}^{n+1})^{t}\end{array}&-(\Delta_{n}^{p})^{(n)}\end{matrix}\right|$
$\displaystyle=$
$\displaystyle(-1)^{(n+1)}|\Gamma_{n+1}^{p}|\cdot|-\Delta_{n-1}^{p}|$
$\displaystyle=$ $\displaystyle|\Gamma_{n+1}^{p}|\cdot|\Delta_{n-1}^{p}|,$

then

$\left|\begin{matrix}\Gamma_{n}^{p}&\Gamma_{n}^{p+1}\\\
\Gamma_{n}^{p+1}&-\Gamma_{n}^{p}\end{matrix}\right|=(-1)^{n}|\Gamma_{n}^{p}|\cdot|\Delta_{n}^{p}|+(-1)^{n+1}|\Gamma_{n+1}^{p}|\cdot|\Delta_{n-1}^{p}|.$

∎

Let

$P(n)=(e_{1},e_{4},\cdots,e_{3n_{1}-2},e_{2},e_{5},\cdots,e_{3n_{2}-1},e_{3},e_{6},\cdots,e_{3n_{3}}),$

where $n_{1}=[\frac{n+2}{3}],n_{2}=[\frac{n+1}{3}],n_{3}=[\frac{n}{3}]$ and
$e_{j}$ is the $j$-th unit column vector of order $n$, i.e., the column vector
with $1$ as the $j$-th entry and zeros elsewhere. And $|P(n)|=\pm 1$. For
simplicity, we write $P$ instead of $P(n)$, when no confusion can occur. When
consider $P(3n),P(3n+1),P(3n+2)$, the following diagram shows $n_{1},n_{2}$
and $n_{3}$ in these cases:

$\begin{array}[]{c|ccc}&n_{1}&n_{2}&n_{3}\\\ \hline\cr 3n&n&n&n\\\
3n+1&n+1&n&n\\\ 3n+2&n+1&n+1&n\end{array}$

Suppose $M=(m_{i,j})_{1\leq i,j\leq n}$ is an $n\times n$ matrix, then

$P^{t}MP=\left(\begin{matrix}(m_{3i-2,3j-2})_{n_{1}\times
n_{1}}&(m_{3i-2,3j-1})_{n_{1}\times n_{2}}&(m_{3i-2,3j})_{n_{1}\times
n_{3}}\\\ (m_{3i-1,3j-2})_{n_{2}\times n_{1}}&(m_{3i-1,3j-1})_{n_{2}\times
n_{2}}&(m_{3i-1,3j})_{n_{2}\times n_{3}}\\\ (m_{3i,3j-2})_{n_{3}\times
n_{1}}&(m_{3i,3j-1})_{n_{3}\times n_{2}}&(m_{3i,3j})_{n_{3}\times
n_{3}}\end{matrix}\right),$

where $(m_{3i-2,3j-1})_{s\times t}$ means the matrix $(m_{3i-2,3j-1})_{1\leq
i\leq s,1\leq j\leq t}$.

When $M=H_{3n}^{p},~{}H_{3n+1}^{p}$ and $H_{3n+2}^{p}$, we have

(2.5) $\displaystyle P^{t}H_{3n}^{p}P$ $\displaystyle=$
$\displaystyle\begin{pmatrix}(U_{p+3(i+j-2)})_{n\times
n}&(U_{p+3(i+j-2)+1})_{n\times n}&(U_{p+3(i+j-2)+2})_{n\times n}\\\
(U_{p+3(i+j-2)+1})_{n\times n}&(U_{p+3(i+j-2)+2})_{n\times
n}&(U_{p+3(i+j-2)+3})_{n\times n}\\\ (U_{p+3(i+j-2)+2})_{n\times
n}&(U_{p+3(i+j-2)+3})_{n\times n}&(U_{p+3(i+j-2)+4})_{n\times n}\end{pmatrix}$
$\displaystyle=$
$\displaystyle\begin{pmatrix}K_{n}^{p}&K_{n}^{p+1}&K_{n}^{p+2}\\\
K_{n}^{p+1}&K_{n}^{p+2}&K_{n}^{p+3}\\\
K_{n}^{p+2}&K_{n}^{p+3}&K_{n}^{p+4}\end{pmatrix},$ (2.6) $\displaystyle
P^{t}H_{3n+1}^{p}P$ $\displaystyle=$
$\displaystyle\begin{pmatrix}K_{n+1}^{p}&(K_{n+1}^{p+1})^{(n+1)}&(K_{n+1}^{p+2})^{(n+1)}\\\
(K_{n+1}^{p+1})_{(n+1)}&K_{n}^{p+2}&K_{n}^{p+3}\\\
(K_{n+1}^{p+2})_{(n+1)}&K_{n}^{p+3}&K_{n}^{p+4}\end{pmatrix},$ (2.7)
$\displaystyle P^{t}H_{3n+2}^{p}P$ $\displaystyle=$
$\displaystyle\begin{pmatrix}K_{n+1}^{p}&K_{n+1}^{p+1}&(K_{n+1}^{p+2})^{(n+1)}\\\
K_{n+1}^{p+1}&K_{n+1}^{p+2}&(K_{n+1}^{p+3})^{(n+1)}\\\
(K_{n+1}^{p+2})_{(n+1)}&(K_{n+1}^{p+3})_{(n+1)}&K_{n}^{p+4}\end{pmatrix},$

##  3\. Recurrent equations

In this section, we establish the recurrence formulae for the sequence
$|\Gamma_{n}^{p}|(n\geq 2,p\geq 0)$, which is the key result in this paper.
Through these eighteen recurrence formulae, we can evaluate all the Hankel
determinants $|\Gamma_{n}^{p}|,|\Delta_{n}^{p}|(n\geq 2,p\geq 0)$.

######  Theorem 1.

For $p\geq 0$ and $n\geq 2$, one has

  1. (1)

$|\Gamma_{3n}^{3p}|=(-1)^{n}|\Gamma_{n}^{p}|^{2}\cdot|\Delta_{n}^{p}|+(-1)^{n+1}|\Gamma_{n}^{p}|\cdot|\Gamma_{n+1}^{p}|\cdot|\Delta_{n-1}^{p}|,$

  2. (2)

$|\Gamma_{3n+1}^{3p}|=(-1)^{n}|\Gamma_{n}^{p}|\cdot|\Gamma_{n+1}^{p}|\cdot|\Delta_{n}^{p}|+(-1)^{n+1}|\Gamma_{n+1}^{p}|^{2}\cdot|\Delta_{n-1}^{p}|,$

  3. (3)

$|\Gamma_{3n+2}^{3p}|=(-1)^{n}|\Gamma_{n+1}^{p}|^{2}\cdot|\Delta_{n}^{p}|,$

  4. (4)

$|\Gamma_{3n}^{3p+1}|=(-1)^{n}|\Gamma_{n}^{p}|\cdot|\Gamma_{n}^{p+1}|\cdot|\Delta_{n}^{p}|+(-1)^{n+1}|\Gamma_{n}^{p+1}|\cdot|\Gamma_{n+1}^{p}|\cdot|\Delta_{n-1}^{p}|,$

  5. (5)

$|\Gamma_{3n+1}^{3p+1}|=(-1)^{n+1}|\Gamma_{n+1}^{p}|^{2}\cdot|\Delta_{n-1}^{p+1}|,$

  6. (6)

$|\Gamma_{3n+2}^{3p+1}|=(-1)^{n+1}|\Gamma_{n+1}^{p}|^{2}\cdot|\Delta_{n}^{p+1}|,$

  7. (7)

$|\Gamma_{3n}^{3p+2}|=(-1)^{n}|\Gamma_{n}^{p+1}|^{2}\cdot|\Delta_{n}^{p}|,$

  8. (8)

$|\Gamma_{3n+1}^{3p+2}|=(-1)^{n}|\Gamma_{n}^{p+1}|\cdot|\Gamma_{n+1}^{p}|\cdot|\Delta_{n}^{p+1}|+(-1)^{n+1}|\Gamma_{n+1}^{p}|\cdot|\Gamma_{n+1}^{p+1}|\cdot|\Delta_{n-1}^{p+1}|,$

  9. (9)

$|\Gamma_{3n+2}^{3p+2}|=(-1)^{n+1}|\Gamma_{n+1}^{p+1}|^{2}\cdot|\Delta_{n}^{p}|,$

  10. (10)

$|\Delta_{3n}^{3p}|\equiv(-1)^{n}|\Gamma_{n}^{p}|\cdot|\Delta_{n}^{p}|^{2}+(-1)^{n+1}|\Gamma_{n+1}^{p}|\cdot|\Delta_{n-1}^{p}|\cdot|\Delta_{n}^{p}|,$

  11. (11)

$|\Delta_{3n+1}^{3p}|\equiv(-1)^{n+1}|\Gamma_{n+1}^{p}|\cdot|\Delta_{n}^{p}|^{2},$

  12. (12)

$|\Delta_{3n+2}^{3p}|\equiv(-1)^{n}|\Gamma_{n+2}^{p}|\cdot|\Delta_{n}^{p}|^{2}+(-1)^{n+1}|\Gamma_{n+1}^{p}|\cdot|\Delta_{n}^{p}|\cdot|\Delta_{n+1}^{p}|,$

  13. (13)

$|\Delta_{3n}^{3p+1}|\equiv(-1)^{n}|\Gamma_{n}^{p+1}|\cdot|\Delta_{n}^{p}|^{2},$

  14. (14)

$|\Delta_{3n+1}^{3p+1}|\equiv(-1)^{n}|\Gamma_{n+1}^{p+1}|\cdot|\Delta_{n}^{p}|^{2},$

  15. (15)

$|\Delta_{3n+2}^{3p+1}|\equiv(-1)^{n}|\Gamma_{n+2}^{p}|\cdot|\Delta_{n}^{p}|\cdot|\Delta_{n}^{p+1}|+(-1)^{n+1}|\Gamma_{n+1}^{p}|\cdot|\Delta_{n}^{p+1}|\cdot|\Delta_{n+1}^{p}|,$

  16. (16)

$|\Delta_{3n}^{3p+2}|\equiv(-1)^{n}|\Gamma_{n}^{p+1}|\cdot|\Delta_{n}^{p}|\cdot|\Delta_{n}^{p+1}|+(-1)^{n+1}|\Gamma_{n+1}^{p+1}|\cdot|\Delta_{n}^{p}|\cdot|\Delta_{n-1}^{p+1}|,$

  17. (17)

$|\Delta_{3n+1}^{3p+2}|\equiv(-1)^{n}|\Gamma_{n+1}^{p}|\cdot|\Delta_{n}^{p+1}|^{2},$

  18. (18)

$|\Delta_{3n+2}^{3p+2}|\equiv(-1)^{n}|\Gamma_{n+2}^{p}|\cdot|\Delta_{n}^{p+1}|^{2}.$

###### Proof.

$1)$ Combine (2.3) and (2.5), we have

$\displaystyle|P^{t}\Gamma_{3n}^{3p}P|$ $\displaystyle=$
$\displaystyle\left|\begin{matrix}\Gamma_{n}^{p}&\mathbf{0}_{n\times
n}&\Gamma_{n}^{p}\\\ \mathbf{0}_{n\times n}&\Gamma_{n}^{p}&\Gamma_{n}^{p+1}\\\
\Gamma_{n}^{p}&\Gamma_{n}^{p+1}&\mathbf{0}_{n\times n}\end{matrix}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{matrix}\Gamma_{n}^{p}&\mathbf{0}_{n\times
n}&\mathbf{0}_{n\times n}\\\ \mathbf{0}_{n\times
n}&\Gamma_{n}^{p}&\Gamma_{n}^{p+1}\\\ \mathbf{0}_{n\times
n}&\Gamma_{n}^{p+1}&-\Gamma_{n}^{p}\end{matrix}\right|.$

Hence

$|\Gamma_{3n}^{3p}|=|P^{t}\Gamma_{3n}^{3p}P|=|\Gamma_{n}^{p}|\cdot\left|\begin{matrix}\Gamma_{n}^{p}&\Gamma_{n}^{p+1}\\\
\Gamma_{n}^{p+1}&-\Gamma_{n}^{p}\end{matrix}\right|.$

By Lemma 1,

$|\Gamma_{3n}^{3p}|=(-1)^{n}|\Gamma_{n}^{p}|^{2}\cdot|\Delta_{n}^{p}|+(-1)^{n+1}|\Gamma_{n}^{p}|\cdot|\Gamma_{n+1}^{p}|\cdot|\Delta_{n-1}^{p}|.$

$2)$ Combine (2.3) and (2.6), we have

$\displaystyle|P^{t}\Gamma_{3n+1}^{3p}P|$ $\displaystyle=$
$\displaystyle\left|\begin{matrix}\Gamma_{n+1}^{p}&\mathbf{0}_{{n+1}\times
n}&(\Gamma_{n+1}^{p})^{(n+1)}\\\
\mathbf{0}_{n\times{n+1}}&\Gamma_{n}^{p}&\Gamma_{n}^{p+1}\\\
(\Gamma_{n+1}^{p})_{(n+1)}&\Gamma_{n}^{p+1}&\mathbf{0}_{n\times
n}\end{matrix}\right|$ $\displaystyle=$
$\displaystyle\left|\begin{matrix}\Gamma_{n+1}^{p}&\mathbf{0}_{{n+1}\times
n}&\mathbf{0}_{{n+1}\times n}\\\
\mathbf{0}_{n\times{n+1}}&\Gamma_{n}^{p}&\Gamma_{n}^{p+1}\\\
\mathbf{0}_{n\times{n+1}}&\Gamma_{n}^{p+1}&-\Gamma_{n}^{p}\end{matrix}\right|.$

By Lemma 1,

$\displaystyle|\Gamma_{3n+1}^{3p}|$ $\displaystyle=$
$\displaystyle|\Gamma_{n+1}^{p}|\cdot\left|\begin{matrix}\Gamma_{n}^{p}&\Gamma_{n}^{p+1}\\\
\Gamma_{n}^{p+1}&-\Gamma_{n}^{p}\end{matrix}\right|$ $\displaystyle=$
$\displaystyle(-1)^{n}|\Gamma_{n}^{p}|\cdot|\Gamma_{n+1}^{p}|\cdot|\Delta_{n}^{p}|+(-1)^{n+1}|\Gamma_{n+1}^{p}|^{2}\cdot|\Delta_{n-1}^{p}|.$

$3)$ Combine (2.3) and (2.7), we have

$\displaystyle|P^{t}\Gamma_{3n+2}^{3p}P|$ $\displaystyle=$
$\displaystyle\left|\begin{matrix}\Gamma_{n+1}^{p}&\mathbf{0}_{(n+1)\times(n+1)}&(\Gamma_{n+1}^{p})^{(n+1)}\\\
\mathbf{0}_{(n+1)\times(n+1)}&\Gamma_{n+1}^{p}&(\Gamma_{n+1}^{p+1})^{(n+1)}\\\
(\Gamma_{n+1}^{p})_{(n+1)}&(\Gamma_{n+1}^{p+1})_{(n+1)}&\mathbf{0}_{n\times
n}\end{matrix}\right|$ $\displaystyle=$
$\displaystyle\left|\begin{matrix}\Gamma_{n+1}^{p}&\mathbf{0}_{(n+1)\times(n+1)}&\mathbf{0}_{(n+1)\times
n}\\\
\mathbf{0}_{(n+1)\times(n+1)}&\Gamma_{n+1}^{p}&(\Gamma_{n+1}^{p+1})^{(n+1)}\\\
(\Gamma_{n+1}^{p})_{(n+1)}&(\Gamma_{n+1}^{p+1})_{(n+1)}&-\Gamma_{n}^{p}\end{matrix}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{matrix}\Gamma_{n+1}^{p}&\mathbf{0}_{(n+1)\times(n+1)}&\mathbf{0}_{(n+1)\times
n}\\\ \mathbf{0}_{(n+1)\times(n+1)}&\Gamma_{n+1}^{p}&\mathbf{0}_{(n+1)\times
n}\\\
(\Gamma_{n+1}^{p})_{(n+1)}&(\Gamma_{n+1}^{p+1})_{(n+1)}&-\Gamma_{n}^{p}-\Gamma_{n}^{p+2}\end{matrix}\right|.$

Therefore,

$|\Gamma_{3n+2}^{3p}|=|\Gamma_{n+1}^{p}|^{2}\cdot|-\Gamma_{n}^{p}-\Gamma_{n}^{p+2}|=(-1)^{n}|\Gamma_{n+1}^{p}|^{2}\cdot|\Delta_{n}^{p}|.$

Formulae $(4)$ to $(18)$ can be proved using similar computation. We state the
proof in the appendix. ∎

Now, we will extend those eighteen recurrent formulae for all $n,p\geq 0$.

######  Proposition 1.

Define $|\Delta_{0}^{p}|=1$ for $p\geq 0,$ and

$|\Gamma_{0}^{p}|=\left\\{\begin{array}[]{cc}2&\text{if }p=0\\\ 1&\text{if
}p\geq
1\end{array}\right.,|\Delta_{-1}^{p}|=\left\\{\begin{array}[]{cc}1&\text{if
}p=0\\\ 0&\text{if }p\geq 1\end{array}\right..$

Then formulae of Theorem 1 holds for $p\geq 0$ and $n\geq 0$.

###### Proof.

Using (2.1), (2.2) and the facts: $|\Gamma_{1}^{p}|=c_{p},$ and
$|\Delta_{1}^{p}|=d_{p}.$ We can check the formulae of Theorem 1 one by one. ∎

##  4\. Periodicity and automaticity properties.

The periodicity and automaticity properties of the Hankel determinants
$|\Gamma_{n}^{p}|$, $|\Delta_{n}^{p}|$ $(n,p\geq 0)$ are discussed in this
section. By the recurrent formulae in Theorem 1 and Proposition 1, to
determine the quantities $\\{|\Gamma_{n}^{p}|\\}_{n\geq 0,p\geq
0},\\{|\Delta_{n}^{p}|\\}_{n\geq 0,p\geq 0}$, we only need to determine the
quantities for $p=0$ and $1$. The following two propositions are devoted to
this purpose.

######  Proposition 2.

With the above notation, we have

(4.1) $|\Gamma_{n}^{0}|\equiv\left\\{\begin{array}[]{cc}1&\text{if }n\equiv
1,2\mod 4\\\ 2&\text{if }n\equiv 3,0\mod
4\end{array}\right.,|\Delta_{n}^{0}|\equiv\left\\{\begin{array}[]{cc}2&\text{if
}n\equiv 1,2\mod 4\\\ 1&\text{if }n\equiv 3,0\mod 4\end{array}\right..$

###### Proof.

We will prove these two assertion simultaneously. For $n=0,1,2,$ the above
equalities can be check directly. Assume that the equalities hold for $n\leq
k.$ By the induction hypothesis, we have for all $n<k,$

$\displaystyle|\Gamma_{n}^{0}|^{2}\equiv|\Delta_{n}^{0}|^{2}\equiv 1,$
$\displaystyle|\Gamma_{n}^{0}|\cdot|\Delta_{n}^{0}|\equiv 2,$
$\displaystyle|\Gamma_{n+1}^{0}|\cdot|\Delta_{n-1}^{0}|\equiv 1,$
$\displaystyle|\Gamma_{n+1}^{0}|+|\Delta_{n-1}^{0}|\equiv|\Delta_{n+1}^{2}|.$

Then if $n=k+1=3l(l\geq 1),$ by Theorem 1 (1) and (10), we have

$\displaystyle|\Gamma_{n}^{0}|$ $\displaystyle=$
$\displaystyle(-1)^{l}|\Gamma_{l}^{0}|^{2}\cdot|\Delta_{l}^{0}|+(-1)^{l+1}|\Gamma_{l}^{0}|\cdot|\Gamma_{l+1}^{0}|\cdot|\Delta_{l-1}^{0}|$
$\displaystyle\equiv$
$\displaystyle(-1)^{l}\left(2|\Gamma_{l}^{0}|-|\Gamma_{l}^{0}|\right)$
$\displaystyle\equiv$ $\displaystyle(-1)^{l}|\Gamma_{l}^{0}|,$
$\displaystyle|\Delta_{n}^{0}|$ $\displaystyle\equiv$
$\displaystyle(-1)^{l}|\Gamma_{l}^{0}|\cdot|\Delta_{l}^{0}|^{2}+(-1)^{l+1}|\Gamma_{l+1}^{0}|\cdot|\Delta_{l-1}^{0}|\cdot|\Delta_{l}^{0}|$
$\displaystyle\equiv$
$\displaystyle(-1)^{l}\left(2|\Delta_{l}^{0}|-|\Delta_{l}^{0}|\right)$
$\displaystyle\equiv$ $\displaystyle(-1)^{l}|\Delta_{l}^{0}|.$

Since $n=3l\equiv-l(\textrm{mod }4)$, the above two equalities implies (4.1).

When $n=k+1=3l+1(l\geq 1),$ by formulae (2) and (11) of Theorem 1,

$\displaystyle|\Gamma_{3l+1}^{0}|$ $\displaystyle=$
$\displaystyle(-1)^{l}|\Gamma_{l}^{0}|\cdot|\Gamma_{l+1}^{0}|\cdot|\Delta_{l}^{0}|+(-1)^{l+1}|\Gamma_{l+2}^{0}|^{2}\cdot|\Delta_{l-1}^{0}|$
$\displaystyle\equiv$
$\displaystyle(-1)^{l}\left(2|\Gamma_{l+1}^{0}|-|\Delta_{l-1}^{0}|\right)$
$\displaystyle\equiv$ $\displaystyle(-1)^{l+1}|\Delta_{l+1}^{0}|,$
$\displaystyle|\Delta_{3l+1}^{0}|$ $\displaystyle\equiv$
$\displaystyle(-1)^{l+1}|\Gamma_{l+1}^{0}|\cdot|\Delta_{l}^{0}|^{2}$
$\displaystyle\equiv$ $\displaystyle(-1)^{l+1}|\Gamma_{l+1}^{0}|.$

Note that $n=3l+1\equiv 1-l(\textrm{mod }4)$, (4.1) follows from the above two
equalities.

When $n=k+1=3l+2(l\geq 1),$ by formulae (3) and (12) of Theorem 1, we have

$\displaystyle|\Gamma_{3l+2}^{0}|$ $\displaystyle=$
$\displaystyle(-1)^{l}|\Gamma_{l+1}^{0}|^{2}\cdot|\Delta_{l}^{0}|$
$\displaystyle\equiv$ $\displaystyle(-1)^{l}|\Delta_{l}^{0}|,$
$\displaystyle|\Delta_{3l+2}^{0}|$ $\displaystyle\equiv$
$\displaystyle(-1)^{l}|\Gamma_{l+2}^{0}|\cdot|\Delta_{l}^{0}|^{2}+(-1)^{l+1}|\Gamma_{l+1}^{0}|\cdot|\Delta_{l}^{0}|\cdot|\Delta_{l+1}^{0}|$
$\displaystyle\equiv$
$\displaystyle(-1)^{l}\left(|\Delta_{l}^{0}|-2|\Delta_{l}^{0}|\right)=(-1)^{l+1}|\Delta_{l}^{0}|.$

These two equalities, combining with the fact $n=3l+2\equiv 2-l(\textrm{mod
}4),$ lead to (4.1). Thus the assertions are proved. ∎

######  Proposition 3.

For $p=1$, we have

(4.2)
$|\Gamma_{n}^{1}|\equiv|\Delta_{n}^{1}|\equiv\left\\{\begin{array}[]{ll}0&\text{if
}n\equiv 1,3\mod 4\\\ 2&\text{if }n\equiv 2\mod 4\\\ 1&\text{if }n\equiv 0\mod
4\end{array}\right..$

###### Proof.

These two assertions will be proved simultaneously. For $n=0,1,2,$ the above
equalities can be check directly. Assume that the equalities hold for $n\leq
k.$ According to 4.1, for all $n\geq 1,$

$|\Gamma_{n}^{0}|^{2}\equiv|\Delta_{n}^{0}|^{2}\equiv
1,|\Gamma_{n}^{0}|\cdot|\Delta_{n}^{0}|\equiv
2,|\Gamma_{n+1}^{0}|\cdot|\Delta_{n-1}^{0}|\equiv 1.$

By the induction hypothesis, for all $1\leq n<k,$

$\left|\Gamma_{n-1}^{1}\right|\equiv-\left|\Gamma_{n+1}^{1}\right|.$

Then if $n=k+1=3l(l\geq 1),$ by formulae (4) and (13) of Theorem 1, we have

$\displaystyle|\Gamma_{n}^{1}|$ $\displaystyle=$
$\displaystyle(-1)^{l}|\Gamma_{l}^{0}|\cdot|\Gamma_{l}^{1}|\cdot|\Delta_{l}^{0}|+(-1)^{l+1}|\Gamma_{l}^{1}|\cdot|\Gamma_{l+1}^{0}|\cdot|\Delta_{l-1}^{0}|$
$\displaystyle\equiv$
$\displaystyle(-1)^{l}\left(2|\Gamma_{l}^{1}|-|\Gamma_{l}^{1}|\right)$
$\displaystyle\equiv$ $\displaystyle(-1)^{l}|\Gamma_{l}^{1}|,$
$\displaystyle|\Delta_{n}^{1}|$ $\displaystyle\equiv$
$\displaystyle(-1)^{l}|\Gamma_{l}^{1}|\cdot|\Delta_{l}^{0}|^{2}\equiv(-1)^{l}|\Gamma_{l}^{1}|.$

Since $n=3l\equiv-l(\textrm{mod }4)$, (4.2) holds in this case.

When $n=k+1=3l+1(l\geq 1),$ by formulae (5) and (14) and Theorem 1, we have

$\displaystyle|\Gamma_{n}^{1}|$ $\displaystyle=$
$\displaystyle(-1)^{l+1}|\Gamma_{l+1}^{0}|^{2}\cdot|\Delta_{l-1}^{1}|\equiv(-1)^{l+1}|\Delta_{l-1}^{1}|\equiv(-1)^{l}|\Gamma_{l+1}^{1}|,$
$\displaystyle|\Delta_{n}^{1}|$ $\displaystyle\equiv$
$\displaystyle(-1)^{l}|\Gamma_{l+1}^{1}|\cdot|\Delta_{l}^{0}|^{2}\equiv(-1)^{l}|\Gamma_{l+1}^{1}|.$

Since $n=3l+1\equiv 1-l(\textrm{mod }4)$, (4.2) holds in this case.

When $n=k+1=3l+2(l\geq 1),$ by formulae (6) and (15) of Theorem 1, we have

$\displaystyle|\Gamma_{n}^{1}|$ $\displaystyle=$
$\displaystyle(-1)^{l+1}|\Gamma_{l+1}^{0}|^{2}\cdot|\Delta_{l}^{1}|\equiv(-1)^{l+1}|\Delta_{l}^{1}|,$
$\displaystyle|\Delta_{n}^{1}|$ $\displaystyle\equiv$
$\displaystyle(-1)^{l}|\Gamma_{l+2}^{0}|\cdot|\Delta_{l}^{0}|\cdot|\Delta_{l}^{1}|+(-1)^{l+1}|\Gamma_{l+1}^{0}|\cdot|\Delta_{l}^{1}|\cdot|\Delta_{l+1}^{0}|$
$\displaystyle\equiv$
$\displaystyle(-1)^{l}\left(|\Delta_{l}^{1}|-2|\Delta_{l}^{1}|\right)\equiv(-1)^{l+1}|\Delta_{l}^{1}|.$

These two equalities, combining with the fact $n=3l+2\equiv 2-l(\textrm{mod
}4),$ lead to (4.2). Thus the assertions are proved. ∎

###  4.1. Periodicity properties.

Let $(u_{n})_{n\geq 0}$ be a sequence with $u_{n}\in\mathbb{F}_{3}$, then the
formal power series

$u(x)=\sum_{n\geq 0}u_{n}x^{n}$

is called the _generating series_ of the sequence $(u_{n})_{n\geq 0}.$

A sequence $(u_{n})_{n\geq 0}$ is periodic of period $t$ if and only if its
generating series adds up to a rational fraction of the form
$\frac{P(x)}{1-x^{t}}$, where $P(x)$ is a polynomial of degree less than $t$.

Let $P(x)=\sum_{n\geq 0}a_{n}x^{n}$ and $Q(x)=\sum_{n\geq 0}b_{n}x^{n}$ be two
formal power series with $a_{n},b_{n}\in\mathbb{F}_{3}$, then their _Hadamard
product_ is defined to be

$P(x)\star Q(x)=\sum_{n\geq 0}a_{n}b_{n}x^{n}.$

In addition, if $(a_{n})_{n\geq 0}$ and $(b_{n})_{n\geq 0}$ are periodic of
period $s$ and $t$ respectively, then $P(x)\star Q(x)$ is the generating
series of the periodic sequence $(a_{n}b_{n})_{n\geq 0}$ having $[s,t]$ as a
period, where $[s,t]$ denotes the lowest common multiple of $s$ and $t$.

For $p\geq 0$, define

$\begin{array}[]{ll}f^{(p)}(x)=\sum_{n\geq
0}\left|\Gamma_{n}^{p}\right|x^{n},&g^{(p)}(x)=\sum_{n\geq
0}\left|D_{n}^{p}\right|x^{n},\end{array}$

and

$h_{0}(x)=\sum_{n\geq 0}(-1)^{n}x^{n},\ h_{1}(x)=\sum_{n\geq
0}(-1)^{n+1}x^{n}$

where the coefficients are taken modulo $3$, with the convention of
Proposition 1.

By Proposition 2 and Proposition 3, we have

(4.3)
$\left\\{\begin{array}[]{ll}f^{(0)}=\frac{2+x+x^{2}+2x^{3}}{1-x^{4}},&g^{(0)}=\frac{1+2x+2x^{2}+x^{3}}{1-x^{4}},\\\
f^{(1)}=\frac{1+2x^{2}}{1-x^{4}},&g^{(1)}=\frac{1+2x^{2}}{1-x^{4}}.\end{array}\right.$

Using the recurrent formulae in Theorem 1, we can compute the above quantities
recursively. We compute $f^{(2)}$ and $g^{(2)}$ as an example.

$\displaystyle f^{(2)}$ $\displaystyle=$ $\displaystyle\sum_{n\geq
0}|\Gamma_{n}^{2}|x^{n}=\sum_{n\geq
0}(|\Gamma_{3n}^{2}|+|\Gamma_{3n+1}^{2}|x+|\Gamma_{3n+2}^{2}|x^{2})x^{3n}$
$\displaystyle=$ $\displaystyle\sum_{n\geq
0}\Big{[}(-1)^{n}|\Gamma_{n}^{1}|^{2}|\Delta_{n}^{0}|+\big{(}(-1)^{n}|\Gamma_{n}^{1}||\Gamma_{n+1}^{0}||\Delta_{n}^{1}|$
$\displaystyle+(-1)^{n+1}|\Gamma_{n+1}^{0}||\Gamma_{n+1}^{1}||\Delta_{n-1}^{1}|\big{)}x+(-1)^{n+1}|\Gamma_{n+1}^{1}|^{2}|\Delta_{n}^{0}|x^{2}\Big{]}x^{3n}$
by Theorem 1 $\displaystyle=$ $\displaystyle(h_{0}\star f^{(1)}\star
f^{(1)}\star g^{(0)})^{3}+x(h_{0}\star f^{(1)}\star\widehat{f^{(0)}}\star
g^{(1)}$
$\displaystyle+h_{1}\star\widehat{f^{(0)}}\star\widehat{f^{(1)}}\star\overline{g^{(1)}})^{3}+x^{2}(h_{1}\star\widehat{f^{(1)}}\star\widehat{f^{(1)}}\star
g^{(0)})^{3}$

where

(4.4) $h_{0}(x)=\sum_{n\geq 0}(-1)^{n}x^{n},\ h_{1}(x)=\sum_{n\geq
0}(-1)^{n+1}x^{n}$

and

(4.5) $\widehat{f^{(1)}}(x)=\sum_{n\geq 0}\left|\Gamma_{n+1}^{1}\right|x^{n},\
\overline{g^{(1)}}(x)=\sum_{n\geq 0}\left|\Delta_{n-1}^{1}\right|x^{n}.$

Thus by (4.3), (4.4) and (4.5), we have

$\displaystyle f^{(2)}$ $\displaystyle=$
$\displaystyle\left(\frac{1+2x^{2}}{1-x^{4}}\right)^{3}+x\left(\frac{1+2x^{2}}{1-x^{4}}+\frac{2x+x^{3}}{1-x^{4}}\right)^{3}+x^{2}\left(\frac{2x+x^{3}}{1-x^{4}}\right)^{3}$
$\displaystyle=$
$\displaystyle\frac{1+x+2x^{4}+2x^{5}+2x^{6}+2x^{7}+x^{10}+x^{11}}{1-x^{12}};$
$\displaystyle g^{(2)}$ $\displaystyle=$ $\displaystyle\sum_{n\geq
0}|\Delta_{n}^{2}|x^{n}=\sum_{n\geq
0}(|\Delta_{3n}^{2}|+x|\Delta_{3n+1}^{2}|+x^{2}|\Delta_{3n+2}^{2}|)x^{3n}$
$\displaystyle=$ $\displaystyle\sum_{n\geq
0}\Big{[}\left((-1)^{n}|\Gamma_{n}^{1}||\Delta_{n}^{0}||\Delta_{n}^{1}|+(-1)^{n+1}|\Gamma_{n+1}^{1}||\Delta_{n}^{0}||\Delta_{n-1}^{1}|\right)$
$\displaystyle+x\big{(}(-1)^{n}|\Gamma_{n+1}^{0}||\Delta_{n}^{1}|^{2}\big{)}+x^{2}\big{(}(-1)^{n}|\Gamma_{n+2}^{0}||\Delta_{n}^{1}|^{2}\big{)}\Big{]}x^{3n}$
$\displaystyle=$ $\displaystyle(h_{0}\star f^{(1)}\star g^{(0)}\star
g^{(1)}+h_{1}\star\widehat{f^{(1)}}\star g^{(0)}\star\overline{g^{(1)}})^{3}$
$\displaystyle+x(h_{0}\star\widehat{f^{(0)}}\star g^{(1)}\star
g^{(1)})^{3}+x^{2}(h_{0}\star\widehat{\widehat{f^{(0)}}}\star g^{(1)}\star
g^{(1)})^{3}$

where

(4.6) $\widehat{\widehat{f^{(0)}}}(x)=\sum_{n\geq
0}\left|\Gamma_{n+2}^{0}\right|x^{n}.$

Thus by (4.3), (4.4), (4.5) and (4.6), we have

$g^{(2)}=\frac{1+x+x^{2}+2x^{6}+2x^{7}+2x^{8}}{1-x^{12}}.$

######  Theorem 2.

For any $p\geq 0$, the sequences $($modulo $3)$

$\\{\left|\Gamma_{n}^{p}\right|\\}_{n\geq
0},\\{\left|\Delta_{n}^{p}\right|\\}_{n\geq 0}$

are both periodic. Moreover, $12\cdot 3^{k}$ is a period if $3^{k}+1\leq p\leq
3^{k+1}.$

###### Proof.

For $p=0,1,2,3$ by the recurrence formulae of Proposition 1 and equalities
(4.3) these two sequences are periodic. Now suppose $p\geq 4$, we shall prove
by induction on $k$ that $12\cdot 3^{k}$ is a period if $3^{k}+1\leq p\leq
3^{k+1}.$

By calculation, we can find that the conclusion is true for $k=1$. Suppose
that the conclusion is true for $p\leq 3^{k}.$ We need to show that the
conclusion is true for $3^{k}+1\leq p\leq 3^{k+1}.$ If $p=3q$, then
$3^{k-1}+1\leq q\leq 3^{k}$, thus by Theorem 1 (1) (2) and (3), we have

$\displaystyle|\Gamma_{3n}^{p}|$ $\displaystyle=$
$\displaystyle(-1)^{n}|\Gamma_{n}^{q}|^{2}\cdot|\Delta_{n}^{q}|+(-1)^{n+1}|\Gamma_{n}^{q}|\cdot|\Gamma_{n+1}^{q}|\cdot|\Delta_{n-1}^{q}|,$
(4.7) $\displaystyle|\Gamma_{3n+1}^{p}|$ $\displaystyle=$
$\displaystyle(-1)^{n}|\Gamma_{n}^{q}|\cdot|\Gamma_{n+1}^{q}|\cdot|\Delta_{n}^{q}|+(-1)^{n+1}|\Gamma_{n+2}^{q}|^{2}\cdot|\Delta_{n-1}^{q}|,$
$\displaystyle|\Gamma_{3n+2}^{p}|$ $\displaystyle=$
$\displaystyle(-1)^{n}|\Gamma_{n+1}^{q}|^{2}\cdot|\Delta_{n}^{q}|.$

By the induction hypothesis, all the sequences occurring on the right hand of
(4.7) have period $12\cdot 3^{k-1}$, and so do the product and sum of these
sequences. Therefore, the sequences
$\left|\Gamma_{3n}^{p}\right|,\left|\Gamma_{3n+1}^{p}\right|$ and
$\left|\Gamma_{3n+2}^{p}\right|$ are all $12\cdot 3^{k-1}$-periodic which
implies that the sequence $\left|\Gamma_{n}^{p}\right|$ is of period $12\cdot
3^{k}$. The cases $p=3q+1$ and $3q+2$ follow in the same way. Similar
discussions can be applied to the sequence $\left|\Delta_{n}^{p}\right|$. ∎

###  4.2. Automaticity properties.

First, we will recall some definitions of two dimensional automatic sequences
which can be found in [2, Chapter 14].

Let $\mathcal{A},\mathcal{B}$ be two finite alphabets. If

$A=(a_{i,j})_{0\leq i\leq m,0\leq j\leq n}$

is an $m\times n$ matrix with entries in $\mathcal{A}$, and
$\psi:\mathcal{A}\rightarrow\mathcal{B}^{k\times l}$ is a _$[k,l]$ -uniform
matrix-valued morphism_, i.e., a map sending each letter in $\mathcal{A}$ to
an $k\times l$ matrix, then $\psi(A)$ is an $km\times ln$ matrix given by

$\left[\begin{array}[]{cccc}\psi(a_{00})&\psi(a_{01})&\cdots&\psi(a_{0,n-1})\\\
\psi(a_{10})&\psi(a_{11})&\cdots&\psi(a_{1,n-1})\\\
\vdots&\vdots&\ddots&\vdots\\\
\psi(a_{m-1,0})&\psi(a_{m-1,1})&\cdots&\psi(a_{m-1,n-1})\\\
\end{array}\right].$

A _$[k,l]$ -automatic sequence_ is the image (under a coding) of a fixed point
of a $[k,l]$-morphism. In particular, if $k=l$, the $[k,k]$-automatic sequence
is also called the $k$-_automatic sequence_. A well known result [2, Theorem
14.2.2](see also [9],[10]) shows that the two-dimensional sequence
$\mathbf{u}=(u_{n,m})_{n,m\geq 0}$ is $[k,l]$-automatic sequences if and only
if the $[k,l]$-kernel $K_{k,l}(\mathbf{u})$ is finite, where

$K_{k,l}(\mathbf{u})=\\{(u_{k^{a}m+r,l^{a}n+s})_{m,n\geq 0}:a\geq 0,0\leq
r<k^{a},0\leq s<l^{a}\\}.$

######  Theorem 3.

The two-dimensional sequences $($modulo $3)$

$\\{\left|\Gamma_{n}^{p}\right|\\}_{n\geq 0,p\geq
0},\\{\left|\Delta_{n}^{p}\right|\\}_{n\geq 0,p\geq 0}$

are both $3$-automatic.

###### Proof.

For this, we only need to show that the $3$-kernels of these sequences are
finite.

Let $\\{u_{n}^{p}\\}_{n\geq-1,p\geq 0}$ be a double sequence. For
$\alpha^{\prime}\in\\{-1,0,1,2\\}$ and
$\alpha,\beta,\beta^{\prime}\in\\{0,1,2\\}$, operations
$S_{\alpha^{\prime}}^{\beta^{\prime}}$ and $T_{\alpha}^{\beta}$ are defined as
follows

$S_{\alpha^{\prime}}^{\beta^{\prime}}u=\\{u_{n+\alpha^{\prime}}^{p+\beta^{\prime}}\\}_{n\geq
0,p\geq 0},\ T_{\alpha}^{\beta}u=\\{u_{3n+\alpha}^{3p+\beta}\\}_{n\geq 0,p\geq
0}.$

Then, for $\alpha^{\prime}\in\\{-1,0,1,2\\}$ and
$\alpha,\beta,\beta^{\prime}\in\\{0,1,2\\}$, we have

(4.8)
$T_{\alpha}^{\beta}S_{\alpha^{\prime}}^{\beta^{\prime}}=\left\\{\begin{array}[]{ll}S_{-1}^{0}T_{2}^{\beta+\beta^{\prime}}&\text{
if }\alpha+\alpha^{\prime}=-1\text{ and }\beta+\beta^{\prime}\leq 2,\\\
S_{-1}^{1}T_{2}^{\beta+\beta^{\prime}-3}&\text{ if
}\alpha+\alpha^{\prime}=-1\text{ and }\beta+\beta^{\prime}\geq 3,\\\
T_{\alpha+\alpha^{\prime}}^{\beta+\beta^{\prime}}&\text{ if
}0\leq\alpha+\alpha^{\prime}\leq 2\text{ and }\beta+\beta^{\prime}\leq 2,\\\
S_{0}^{1}T_{\alpha+\alpha^{\prime}}^{\beta+\beta^{\prime}-3}&\text{ if
}0\leq\alpha+\alpha^{\prime}\leq 2\text{ and }\beta+\beta^{\prime}\geq 3,\\\
S_{1}^{0}T_{\alpha+\alpha^{\prime}-3}^{\beta+\beta^{\prime}}&\text{ if
}\alpha+\alpha^{\prime}\geq 3\text{ and }\beta+\beta^{\prime}\leq 2,\\\
S_{1}^{1}T_{\alpha+\alpha^{\prime}-3}^{\beta+\beta^{\prime}-3}&\text{ if
}\alpha+\alpha^{\prime}\geq 3\text{ and }\beta+\beta^{\prime}\geq
3.\end{array}\right.$

Suppose $\Gamma,\Delta$ and $F$ stand for the sequences
$\\{\left|\Gamma_{n}^{p}\right|\\}_{n\geq 0,p\geq
0},\\{\left|\Delta_{n}^{p}\right|\\}_{n\geq 0,p\geq 0}$ and
$\\{F_{n}^{p}\\}_{n\geq 0,p\geq 0}$ modulo $3$ where $F_{n}^{p}=(-1)^{n}.$
Thus for any $\beta\in\\{0,1,2\\}$

(4.9) $T_{0}^{\beta}F=T_{2}^{\beta}F=S_{2}^{\beta}F=F\text{ and
}T_{1}^{\beta}F=S_{1}^{\beta}F=S_{1}^{0}F.$

We rewrite Theorem 1 and Proposition 1 as follows

(4.10) $\left\\{\begin{array}[]{ll}T_{0}^{0}\Gamma\equiv
F\cdot\Gamma^{2}\cdot\Delta&T_{0}^{2}\Gamma\equiv
F\cdot\left(S_{0}^{1}\Gamma\right)^{2}\cdot\Delta,\\\
\lx@intercol\hfil+S_{1}^{0}F\cdot\Gamma\cdot S_{1}^{0}\Gamma\cdot
S_{-1}^{0}\Delta,\lx@intercol&\lx@intercol\hfil\lx@intercol\\\
T_{0}^{0}\Delta\equiv F\cdot\Gamma\cdot\Delta^{2}&T_{0}^{2}\Delta\equiv F\cdot
S_{0}^{1}\Gamma\cdot\Delta\cdot S_{0}^{1}\Delta\\\
\lx@intercol\hfil+S_{1}^{0}F\cdot S_{1}^{0}\Gamma\cdot
S_{-1}^{0}\Delta\cdot\Delta,\lx@intercol&\lx@intercol\hfil+S_{1}^{0}F\cdot
S_{1}^{1}\Gamma\cdot\Delta\cdot S_{-1}^{1}\Delta,\lx@intercol\\\
T_{1}^{0}\Gamma\equiv F\cdot
S_{1}^{0}\Gamma\cdot\Gamma\cdot\Delta&T_{1}^{2}\Gamma\equiv F\cdot
S_{1}^{0}\Gamma\cdot S_{0}^{1}\Gamma\cdot S_{0}^{1}\Delta\\\
\lx@intercol\hfil+S_{1}^{0}F\cdot(S_{1}^{0}\Gamma)^{2}\cdot
S_{-1}^{0}\Delta,\lx@intercol&\lx@intercol\hfil+S_{1}^{0}F\cdot
S_{1}^{0}\Gamma\cdot S_{1}^{1}\Gamma\cdot S_{-1}^{1}\Delta,\lx@intercol\\\
T_{1}^{0}\Delta\equiv S_{1}^{0}F\cdot
S_{1}^{0}\Gamma\cdot\Delta^{2},&T_{1}^{2}\Delta\equiv F\cdot
S_{1}^{0}\Gamma\cdot\left(S_{0}^{1}\Delta\right)^{2},\\\ T_{2}^{0}\Gamma\equiv
F\cdot(S_{1}^{0}\Gamma)^{2}\cdot\Delta,&T_{2}^{2}\Gamma\equiv
S_{1}^{0}F\cdot\left(S_{1}^{1}\Gamma\right)^{2}\cdot\Delta,\\\
T_{2}^{0}\Delta\equiv F\cdot
S_{2}^{0}\Gamma\cdot\Delta^{2}&T_{2}^{2}\Delta\equiv F\cdot
S_{2}^{0}\Gamma\cdot\left(S_{0}^{1}\Delta\right)^{2},\\\
\lx@intercol\hfil+S_{1}^{0}F\cdot S_{1}^{0}\Gamma\cdot\Delta\cdot
S_{1}^{0}\Delta,\lx@intercol&\lx@intercol\hfil\lx@intercol\\\
T_{0}^{1}\Gamma\equiv F\cdot
S_{0}^{1}\Gamma\cdot\Gamma\cdot\Delta&T_{2}^{1}\Gamma\equiv
S_{1}^{0}F\cdot\left(S_{1}^{0}\Gamma\right)^{2}\cdot S_{0}^{1}\Delta,\\\
\lx@intercol\hfil+S_{1}^{0}F\cdot S_{1}^{0}\Gamma\cdot S_{0}^{1}\Gamma\cdot
S_{-1}^{0}\Delta,\lx@intercol&\lx@intercol\hfil\lx@intercol\\\
T_{0}^{1}\Delta\equiv F\cdot
S_{0}^{1}\Gamma\cdot\Delta^{2},&T_{2}^{1}\Delta\equiv F\cdot
S_{2}^{0}\Gamma\cdot\Delta\cdot S_{0}^{1}\Delta\\\
\lx@intercol\hfil\lx@intercol&\lx@intercol\hfil+S_{1}^{0}F\cdot
S_{1}^{0}\Gamma\cdot S_{0}^{1}\Delta\cdot S_{1}^{0}\Delta,\lx@intercol\\\
T_{1}^{1}\Gamma\equiv S_{1}^{0}F\cdot\left(S_{1}^{0}\Gamma\right)^{2}\cdot
S_{-1}^{1}\Delta,&T_{1}^{1}\Delta\equiv F\cdot
S_{1}^{1}\Gamma\cdot\Delta^{2}.\end{array}\right.$

Let $\mathcal{X}=\\{\Gamma,\Delta,F\\}$ and
$\mathcal{Y}=\\{S_{\alpha}^{\beta}J~{}|~{}J\in\mathcal{X},\alpha=-1,0,1,2\text{
and }\beta=0,1,2\\}.$ According to (4.8) (4.9) and (4.10), for any
$\alpha,\beta\in\\{0,1,2\\}$ and $J\in\mathcal{Y}$, $T_{\alpha}^{\beta}J$ can
be expressed as a polynomial with coefficients in $GF_{3}$ of the elements of
$\mathcal{Y}$. Hence the elements of $3$-kernels $K_{3}(\Gamma)$ and
$K_{3}(\Delta)$ are obtained by successive applications of operators
$T_{\alpha}^{\beta}$. For instance, let
$(|\Gamma_{3^{m}n+r}^{3^{m}p+s}|)_{n,p\geq 0}\in K_{3}(\Gamma)$ where $m\geq
0$ and $0\geq r,s\geq 3^{m}$. Suppose $r=\sum_{i=0}^{m-1}3^{i}\alpha_{i}$ and
$s=\sum_{i=0}^{m-1}3^{i}\beta_{i}$ where $\alpha_{i},\beta_{i}\in\\{0,1,2\\}$.
It is easy to verify that $(|\Gamma_{3^{m}n+r}^{3^{m}p+s}|)_{n,p\geq
0}=T_{\alpha_{m-1}}^{\beta_{m-1}}\cdots
T_{\alpha_{1}}^{\beta_{1}}T_{\alpha_{0}}^{\beta_{0}}(\Gamma)$. Therefore the
elements of these two $3$-kernels are polynomials with coefficients in
$GF_{3}$ of the elements of $\mathcal{Y}$.

Since there is only a finite number of polynomial functions on $GF_{3}$ with
twelve variables, these $3$-kernels are finite. Therefore, the sequences
$\\{\left|\Gamma_{n}^{p}\right|\\}_{n\geq 0,p\geq 0}$ and
$\\{\left|\Delta_{n}^{p}\right|\\}_{n\geq 0,p\geq 0}$ are $3$-automatic. ∎

######  Corollary 1.

For any $n\geq 1$, the sequences $($modulo $3)$

$\\{\left|\Gamma_{n}^{p}\right|\\}_{p\geq
0},\\{\left|\Delta_{n}^{p}\right|\\}_{p\geq 0}$

are both $3$-automatic.

###### Proof.

An immediate consequence of Salon [9], [10] is that, if a two-dimensional
sequence $\\{s_{m,n}\\}_{m,n\geq 0}$ is 3-automatic, then for any fixed $m\geq
0$ the sequence $\\{s_{m,n}\\}_{n\geq 0}$ is 3-automatic which prove our
result. ∎

##  5\. Applications.

###  5.1. Padé approximation

Now, consider again the Cantor sequence

$\boldsymbol{c}=c_{0}c_{1}c_{2}\cdots\in\\{0,1\\}^{\mathbb{N}},$

let

$f(x)=\sum_{n\geq 0}c_{n}x^{n}$

be the generating function of the Cantor sequence. It follows from (2.1) that

(5.1) $f(x)=(1+x^{2})f(x^{3}),$

and $f(x)>1$ for any $x>0$.

Denote by $\left[\frac{p}{q}\right]_{f}$, a $(p,q)$-order Padé approximate of
$f$, i.e., a rational function $P(x)/Q(x)$ whose denominator has degree $q$
and whose numerator has degree $p$ such that

$f(x)-\frac{P(x)}{Q(x)}=O(x^{p+q+1}),\ x\longrightarrow 0.$

A classical result [4, Brezinski, Page 35], shows us that if
$\Gamma_{n}^{0}\neq 0$, then the Padé approximate
$\left[\frac{n-1}{n}\right]_{f}$ exists. Moreover,

(5.2)
$f(x)-\left[n-1/n\right]_{f}(x)=\frac{\Gamma_{n+1}^{0}}{\Gamma_{n}^{0}}x^{2n}+O(x^{2n+1}).$

Hence by Proposition 2, we have the following theorem.

######  Theorem 4.

Let $f(x)=\sum_{n\geq 0}c_{n}x^{n}$, then for any $n\geq 1$, the
$(n-1,n)$-order Padé approximate of $f$ exists.

###  5.2. The irrationality exponent of the Cantor number

Let $\xi$ be an irrational number. The _irrationality exponent_ (or
_irrational measure_) $\mu(\xi)$ of $\xi$ is defined as follow

$\mu(\xi)=\sup\left\\{\mu\in\mathbb{R}:\left|\xi-\frac{p}{q}\right|<\frac{1}{q^{\mu}}\text{
for infinite many }(p,q)\in\mathbb{Z}\times\mathbb{N}\right\\}.$

Let $\xi_{\mathbf{c},b}$ be the _Cantor number_ defined by

$\xi_{\mathbf{c},b}=\sum_{k\geq
0}\frac{c_{k}}{b^{k}}=1+\frac{1}{b^{2}}+\frac{1}{b^{6}}+\frac{1}{b^{8}}+\cdots,$

where $\mathbf{c}=c_{0}c_{1}c_{2}\cdots$ is the Cantor sequence. Combine
equation (5.1), Propsition 1 and Theorem 1 in [7], we have

######  Proposition 4.

For any integer $b\geq 2$, the irrationality exponent of the Cantor number
$\xi_{\boldsymbol{c},b}$ is equal to $2$.

######  Corollary 2.

For any $b\geq 2,$ let $\eta_{\mathbf{d},b}=\sum_{n\geq 0}d_{n}b^{-n}$. The
irrationality exponent of $\eta_{\mathbf{d},b}$ is equal to $2$.

###### Proof.

Using (2.2), we have

$\displaystyle\eta_{\mathbf{d},b}$ $\displaystyle=$ $\displaystyle\sum_{n\geq
0}d_{n}b^{-n}=\sum_{n\geq 0}(d_{3n}+d_{3n+1}b^{-1}+d_{3n+2}b^{-2})b^{-3n}$
$\displaystyle=$ $\displaystyle\sum_{n\geq
0}(2c_{n}+c_{n+1}b^{-1}+c_{n}b^{-2})b^{-3n}$ $\displaystyle=$ $\displaystyle
2\xi_{\boldsymbol{c},b^{3}}+b^{-1}(\xi_{\boldsymbol{c},b^{3}}-1)+b^{-2}\xi_{\boldsymbol{c},b^{3}}$
$\displaystyle=$
$\displaystyle(2+b^{-1}+b^{-2})\xi_{\boldsymbol{c},b^{3}}-b^{-1}.$

Since the irrationality exponent is invariant under multiplication and
addition of a rational number, we can deduce from Proposition 4 that the
irrationality exponent of $\eta_{\mathbf{d},b}$ is equal to $2.$ ∎

## References

  * [1] J. P. Allouche, J. Peyrière, Z. X. Wen and Z. Y. Wen, Hankel determinants of the Thue-Morse sequence, Ann. Inst. Fourier, Grenoble, 48, 1 (1998), 1-27.
  * [2] J. P. Allouche and J. Shallit, Automatic sequences: Theroy, Applications, Generalizations, Cambridge University Press 2003.
  * [3] G. A. Jr. Baker and P. Graves-Morris, Padé approximations, second edition. Cambridge Unviersity Press, 1996.
  * [4] C. Brezinski, Padé type Approximation and General Orthogonal Polynomials, International Series of Numerical Mathematics. Vol. 50, Birkhäuser Verlag, 1980.
  * [5] F. M. Dekking, Recurrent sets, Advance in Mathematics, 44, 78-104 (1982).
  * [6] P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Vol. 1794 of Lecture Notes in Mathematics, Springer-Verlag, 2001.
  * [7] Y.-J. Guo, Z.-X. Wen and W. Wu, On the irrationality exponent of the regular paperfolding numbers, Linear Algebra Appl 446 (2014), 237-446.
  * [8] T. Kamae, J. Tamura and Z.-Y. Wen, Hankel determinants for the Fibonacci word and Padé approximation, Acta Arithmetica 89.2 (1999) 123-161.
  * [9] O. Salon, Suites automatiques à multi-indices et algébricité, C.R. Acad. Sci. Paris, Série I, 305 (1987), 501-504.
  * [10] O. Salon, Suites automatiques à multi-indices, Séminaire de Théorie des Nombres de Bordeaux, Exposé 4, (1986-1987), 4-01-4-27; followed by an appendix by J. Shallit, 4-29A-4-36A.
  * [11] J. A. Shohat and J. D. Tamarkin, The Problem of Moments, New York: American mathematical society (1943).
  * [12] J. Tamura, Padé approximation for the infinite words generated by certain substitutions and Hankel determinants, Number Theory and Its Applications, Kluwer Academic Publishers. (1990), 309-346.

##  Appendix A Proof of Theorem 1 : Continue.

###### Proof.

$4)$ Combine (2.3) and (2.5), we have

$\displaystyle|P^{t}\Gamma_{3n}^{3p+1}P|$ $\displaystyle=$
$\displaystyle\left|\begin{matrix}\mathbf{0}_{n\times
n}&\Gamma_{n}^{p}&\Gamma_{n}^{p+1}\\\
\Gamma_{n}^{p}&\Gamma_{n}^{p+1}&\mathbf{0}_{n\times n}\\\
\Gamma_{n}^{p+1}&\mathbf{0}_{n\times n}&\Gamma_{n}^{p+1}\end{matrix}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{matrix}-\Gamma_{n}^{p+1}&\Gamma_{n}^{p}&\mathbf{0}_{n\times
n}\\\ \Gamma_{n}^{p}&\Gamma_{n}^{p+1}&\mathbf{0}_{n\times n}\\\
\mathbf{0}_{n\times n}&\mathbf{0}_{n\times
n}&\Gamma_{n}^{p+1}\end{matrix}\right|.$

Since

$\left|\begin{matrix}-\Gamma_{n}^{p+1}&\Gamma_{n}^{p}\\\
\Gamma_{n}^{p}&\Gamma_{n}^{p+1}\end{matrix}\right|=\left|\begin{matrix}\Gamma_{n}^{p}&\Gamma_{n}^{p+1}\\\
\Gamma_{n}^{p+1}&-\Gamma_{n}^{p}\end{matrix}\right|,$

by Lemma 1,

$\displaystyle|\Gamma_{3n}^{3p+1}|$ $\displaystyle=$
$\displaystyle|\Gamma_{n}^{p+1}|\cdot\left|\begin{matrix}\Gamma_{n}^{p}&\Gamma_{n}^{p+1}\\\
\Gamma_{n}^{p+1}&-\Gamma_{n}^{p}\end{matrix}\right|$ $\displaystyle=$
$\displaystyle(-1)^{n}|\Gamma_{n}^{p+1}|\cdot|\Gamma_{n}^{p}|\cdot|\Delta_{n}^{p}|+(-1)^{n+1}|\Gamma_{n}^{p+1}|\cdot|\Gamma_{n+1}^{p}|\cdot|\Delta_{n-1}^{p}|.$

$5)$ Combine (2.3) and (2.6), we have

$|P^{t}\Gamma_{3n+1}^{3p+1}P|=\left|\begin{array}[]{ccc}\mathbf{0}_{(n+1)\times(n+1)}&(\Gamma_{n+1}^{p})^{(n+1)}&(\Gamma_{n+1}^{p+1})^{(n+1)}\\\
(\Gamma_{n+1}^{p})_{(n+1)}&\Gamma_{n}^{p+1}&\mathbf{0}_{n\times n}\\\
(\Gamma_{n+1}^{p+1})_{(n+1)}&\mathbf{0}_{n\times
n}&\Gamma_{n}^{p+1}\end{array}\right|.$

Recall that $\alpha_{p}^{n}$ is the column vector of the form
$(c_{p},c_{p+1},\cdots,c_{p+n-1})^{t}$, then

$\displaystyle|P^{t}\Gamma_{3n+1}^{3p+1}P|$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccc}\mathbf{0}_{(n+1)\times(n+1)}&(\Gamma_{n+1}^{p})^{(n+1)}&(\Gamma_{n+1}^{p+1})^{(n+1)}\\\
(\Gamma_{n+1}^{p})_{(n+1)}&\Gamma_{n}^{p+1}&\mathbf{0}_{n\times n}\\\
\mathbf{0}_{(n-1)\times(n+1)}&-(\Gamma_{n}^{p+1})_{(1)}&(\Gamma_{n}^{p+1})_{(n)}\\\
(\alpha_{p+n}^{n+1})^{t}&\mathbf{0}_{1\times
n}&(\alpha_{p+n}^{n})^{t}\end{array}\right|$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{c}(\Gamma_{n+1}^{p})_{(n+1)}\\\
(\alpha_{p+n}^{n+1})^{t}\end{array}\right|\cdot\left|\begin{array}[]{cc}(\Gamma_{n+1}^{p})^{(n+1)}&(\Gamma_{n+1}^{p+1})^{(n+1)}\\\
-(\Gamma_{n}^{p+1})_{(1)}&(\Gamma_{n}^{p+1})_{(n)}\end{array}\right|$
$\displaystyle=$
$\displaystyle\left|\Gamma_{n+1}^{p}\right|\cdot\left|\begin{array}[]{cc}(\Gamma_{n+1}^{p})^{(n+1)}&(\Gamma_{n+1}^{p+1})^{(n+1)}\\\
-(\Gamma_{n}^{p+1})_{(1)}&(\Gamma_{n}^{p+1})_{(n)}\end{array}\right|$
$\displaystyle=$
$\displaystyle\left|\Gamma_{n+1}^{p}\right|\cdot\left|\begin{array}[]{ccc}(\Gamma_{n+1}^{p})^{(n+1)}&0_{(n+1)\times(n-1)}&\alpha_{p+n}^{n+1}\\\
-(\Gamma_{n}^{p+1})_{(1)}&\Gamma_{n-1}^{p+1}+\Gamma_{n-1}^{p+3}&\alpha_{p+n}^{n-1}\end{array}\right|$
$\displaystyle=$
$\displaystyle\left|\Gamma_{n+1}^{p}\right|\cdot(-1)^{n+1}\cdot\left|\begin{array}[]{cc}(\Gamma_{n+1}^{p})^{(n+1)}&\alpha_{p+n}^{n+1}\end{array}\right|\cdot\left|\Gamma_{n-1}^{p+1}+\Gamma_{n-1}^{p+3}\right|$
$\displaystyle=$
$\displaystyle(-1)^{n+1}\left|\Gamma_{n+1}^{p}\right|^{2}\cdot|\Gamma_{n-1}^{p+1}+\Gamma_{n-1}^{p+3}|.$

Therefore,

$|\Gamma_{3n+1}^{3p+1}|=(-1)^{n+1}|\Gamma_{n+1}^{p}|^{2}\cdot|\Delta_{n-1}^{p+1}|.$

$6)$ Combine (2.3) and (2.7), we have

$\displaystyle|P^{t}\Gamma_{3n+2}^{3p+1}P|$ $\displaystyle=$
$\displaystyle\left|\begin{matrix}\mathbf{0}_{(n+1)\times(n+1)}&\Gamma_{n+1}^{p}&(\Gamma_{n+1}^{p+1})^{(n+1)}\\\
\Gamma_{n+1}^{p}&\Gamma_{n+1}^{p+1}&\mathbf{0}_{(n+1)\times n}\\\
(\Gamma_{n+1}^{p+1})_{(n+1)}&\mathbf{0}_{n\times(n+1)}&\Gamma_{n}^{p+1}\end{matrix}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{matrix}\mathbf{0}_{(n+1)\times(n+1)}&\Gamma_{n+1}^{p}&\mathbf{0}_{(n+1)\times
n}\\\ \Gamma_{n+1}^{p}&\Gamma_{n+1}^{p+1}&-(\Gamma_{n+1}^{p+1})^{(1)}\\\
(\Gamma_{n+1}^{p+1})_{(n+1)}&\mathbf{0}_{n\times(n+1)}&\Gamma_{n}^{p+1}\end{matrix}\right|$
$\displaystyle=$
$\displaystyle(-1)^{n+1}\left|\Gamma_{n+1}^{p}\right|\cdot\left|\begin{matrix}\Gamma_{n+1}^{p}&-(\Gamma_{n+1}^{p+1})^{(1)}\\\
(\Gamma_{n+1}^{p+1})_{(n+1)}&\Gamma_{n}^{p+1}\end{matrix}\right|$
$\displaystyle=$
$\displaystyle(-1)^{n+1}\left|\Gamma_{n+1}^{p}\right|\cdot\left|\begin{matrix}\Gamma_{n+1}^{p}&-(\Gamma_{n+1}^{p+1})^{(1)}\\\
\mathbf{0}_{n\times(n+1)}&\Gamma_{n}^{p+1}+\Gamma_{n}^{p+3}\end{matrix}\right|$
$\displaystyle=$
$\displaystyle(-1)^{n+1}\left|\Gamma_{n+1}^{p}\right|^{2}\cdot\left|\Delta_{n}^{p+1}\right|.$

Therefore,

$|\Gamma_{3n++2}^{3p+1}|=(-1)^{n+1}\left|\Gamma_{n+1}^{p}\right|^{2}\cdot\left|\Delta_{n}^{p+1}\right|.$

$7)$ Combine (2.3) and (2.5), we have

$\displaystyle|P^{t}\Gamma_{3n}^{3p+2}P|$ $\displaystyle=$
$\displaystyle\left|\begin{matrix}\Gamma_{n}^{p}&\Gamma_{n}^{p+1}&\mathbf{0}_{n\times
n}\\\ \Gamma_{n}^{p+1}&\mathbf{0}_{n\times n}&\Gamma_{n}^{p+1}\\\
\mathbf{0}_{n\times n}&\Gamma_{n}^{p+1}&\Gamma_{n}^{p+2}\end{matrix}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{matrix}\Gamma_{n}^{p}&\Gamma_{n}^{p+1}&\mathbf{0}_{n\times
n}\\\ \mathbf{0}_{n\times n}&\mathbf{0}_{n\times n}&\Gamma_{n}^{p+1}\\\
-\Gamma_{n}^{p+2}&\Gamma_{n}^{p+1}&\Gamma_{n}^{p+2}\end{matrix}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{matrix}\Gamma_{n}^{p}&\Gamma_{n}^{p+1}&\mathbf{0}_{n\times
n}\\\ \mathbf{0}_{n\times n}&\mathbf{0}_{n\times n}&\Gamma_{n}^{p+1}\\\
-\Gamma_{n}^{p+2}-\Gamma_{n}^{p}&\mathbf{0}_{n\times
n}&\Gamma_{n}^{p+2}\end{matrix}\right|$ $\displaystyle=$
$\displaystyle(-1)^{n}|\Gamma_{n}^{p+1}|^{2}\cdot|\Gamma_{n}^{p}+\Gamma_{n}^{p+2}|.$

Therefore,

$|\Gamma_{3n}^{3p+2}|=(-1)^{n}|\Gamma_{n}^{p+1}|^{2}\cdot|\Delta_{n}^{p}|.$

$8)$ Combine (2.3) and (2.6), we have

$\displaystyle|P^{t}\Gamma_{3n+1}^{3p+2}P|$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccc}\Gamma_{n+1}^{p}&(\Gamma_{n+1}^{p+1})^{(n+1)}&\mathbf{0}_{(n+1)\times
n}\\\ (\Gamma_{n+1}^{p+1})_{(n+1)}&\mathbf{0}_{n\times n}&\Gamma_{n}^{p+1}\\\
\mathbf{0}_{n\times(n+1)}&\Gamma_{n}^{p+1}&\Gamma_{n}^{p+2}\end{array}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccc}\Gamma_{n+1}^{p}&\mathbf{0}_{(n+1)\times
n}&\mathbf{0}_{(n+1)\times n}\\\
(\Gamma_{n+1}^{p+1})_{(n+1)}&-\Gamma_{n}^{p+2}&\Gamma_{n}^{p+1}\\\
\mathbf{0}_{n\times(n+1)}&\Gamma_{n}^{p+1}&\Gamma_{n}^{p+2}\end{array}\right|$
$\displaystyle=$
$\displaystyle|\Gamma_{n+1}^{p}|\cdot\left|\begin{array}[]{cc}-\Gamma_{n}^{p+2}&\Gamma_{n}^{p+1}\\\
\Gamma_{n}^{p+1}&\Gamma_{n}^{p+2}\end{array}\right|$ $\displaystyle=$
$\displaystyle|\Gamma_{n+1}^{p}|\cdot\left|\begin{array}[]{c}\Gamma_{n}^{p+1}\Gamma_{n}^{p+2}\\\
\Gamma_{n}^{p+2}-\Gamma_{n}^{p+1}\end{array}\right|.$

By Lemma 1,

$|\Gamma_{3n+1}^{3p+2}|=(-1)^{n}|\Gamma_{n+1}^{p}|\cdot|\Gamma_{n}^{p+1}|\cdot|\Delta_{n}^{p+1}|+(-1)^{n+1}|\Gamma_{n+1}^{p}|\cdot|\Gamma_{n+1}^{p+1}|\cdot|\Delta_{n-1}^{p+1}|.$

$9)$ Combine (2.3) and (2.7), we have

$\displaystyle|P^{t}\Gamma_{3n+2}^{3p+2}P|$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccc}\Gamma_{n+1}^{p}&\Gamma_{n+1}^{p+1}&\mathbf{0}_{(n+1)\times
n}\\\
\Gamma_{n+1}^{p+1}&\mathbf{0}_{(n+1)\times(n+1)}&(\Gamma_{n+1}^{p+1})^{(n+1)}\\\
\mathbf{0}_{n\times(n+1)}&(\Gamma_{n+1}^{p+1})_{(n+1)}&\Gamma_{n}^{p+2}\end{array}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccc}\Gamma_{n+1}^{p}&\Gamma_{n+1}^{p+1}&-(\Gamma_{n+1}^{p})^{(n+1)}\\\
(\Gamma_{n+1}^{p+1})_{(n+1)}&\mathbf{0}_{(n+1)\times(n+1)}&\mathbf{0}_{(n+1)\times
n}\\\
\mathbf{0}_{n\times(n+1)}&\Gamma_{n}^{p+1}&\Gamma_{n}^{p+2}\end{array}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccc}\Gamma_{n+1}^{p}&\Gamma_{n+1}^{p+1}&-(\Gamma_{n+1}^{p})^{(n+1)}\\\
(\Gamma_{n+1}^{p+1})_{(n+1)}&\mathbf{0}_{(n+1)\times(n+1)}&\mathbf{0}_{(n+1)\times
n}\\\
-(\Gamma_{n+1}^{p})_{(n+1)}&\mathbf{0}_{n\times(n+1)}&\Gamma_{n}^{p}+\Gamma_{n}^{p+2}\end{array}\right|$
$\displaystyle=$
$\displaystyle(-1)^{n+1}|\Gamma_{n+1}^{p+1}|\cdot\left|\begin{array}[]{cc}\Gamma_{n+1}^{p+1}&-(\Gamma_{n+1}^{p})^{(n+1)}\\\
\mathbf{0}_{n\times(n+1)}&\Gamma_{n}^{p}+\Gamma_{n}^{p+2}\end{array}\right|$
$\displaystyle=$
$\displaystyle(-1)^{n+1}|\Gamma_{n+1}^{p+1}|^{2}\cdot|\Gamma_{n}^{p}+\Gamma_{n}^{p+2}|.$

Therefore,

$|\Gamma_{3n+2}^{3p+2}|=(-1)^{n+1}|\Gamma_{n+1}^{p+1}|^{2}\cdot|\Delta_{n}^{p}|.$

$10)$ Combine (2.4) and (2.5), we have

$\displaystyle|P^{t}\Delta_{3n}^{3p}P|$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccc}2\Gamma_{n}^{p}&\Gamma_{n}^{p+1}&\Gamma_{n}^{p}\\\
\Gamma_{n}^{p+1}&\Gamma_{n}^{p}&2\Gamma_{n}^{p+1}\\\
\Gamma_{n}^{p}&2\Gamma_{n}^{p+1}&\Gamma_{n}^{p+2}\end{array}\right|$
$\displaystyle\equiv$
$\displaystyle\left|\begin{array}[]{ccc}-\Gamma_{n}^{p}&\Gamma_{n}^{p+1}&\Gamma_{n}^{p}\\\
\Gamma_{n}^{p+1}&\Gamma_{n}^{p}&-\Gamma_{n}^{p+1}\\\
\Gamma_{n}^{p}&-\Gamma_{n}^{p+1}&\Gamma_{n}^{p+2}\end{array}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccc}-\Gamma_{n}^{p}&\Gamma_{n}^{p+1}&\boldsymbol{0}_{n\times
n}\\\ \Gamma_{n}^{p+1}&\Gamma_{n}^{p}&\boldsymbol{0}_{n\times n}\\\
\Gamma_{n}^{p}&-\Gamma_{n}^{p+1}&\Gamma_{n}^{p}+\Gamma_{n}^{p+2}\end{array}\right|.$

Hence, by Lemma 1,

$\displaystyle|\Delta_{3n}^{3p}|$ $\displaystyle\equiv$
$\displaystyle|\Gamma_{n}^{p}+\Gamma_{n}^{p+2}|\cdot\left|\begin{array}[]{cc}-\Gamma_{n}^{p}&\Gamma_{n}^{p+1}\\\
\Gamma_{n}^{p+1}&\Gamma_{n}^{p}\end{array}\right|$ $\displaystyle=$
$\displaystyle(-1)^{n}|\Gamma_{n}^{p}|\cdot|\Delta_{n}^{p}|^{2}+(-1)^{n+1}|\Gamma_{n+1}^{p}|\cdot|\Delta_{n-1}^{p}|\cdot|\Delta_{n}^{p}|.$

$11)$ Combine (2.4) and (2.6), we have

$\displaystyle\left|P^{t}\Delta_{3n+1}^{3p}P\right|$ $\displaystyle\equiv$
$\displaystyle\left|\begin{array}[]{ccc}-\Gamma_{n+1}^{p}&\left(\Gamma_{n+1}^{p+1}\right)^{(n+1)}&\left(\Gamma_{n+1}^{p}\right)^{(n+1)}\\\
\left(\Gamma_{n+1}^{p+1}\right)_{(n+1)}&\Gamma_{n}^{p}&-\Gamma_{n}^{p+1}\\\
\left(\Gamma_{n+1}^{p}\right)_{(n+1)}&-\Gamma_{n}^{p+1}&\Gamma_{n}^{p+2}\end{array}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccc}-\Gamma_{n+1}^{p}&\left(\Gamma_{n+1}^{p+1}\right)^{(n+1)}&\boldsymbol{0}_{(n+1)\times
n}\\\
\left(\Gamma_{n+1}^{p+1}\right)_{(n+1)}&\Gamma_{n}^{p}&\boldsymbol{0}_{n\times
n}\\\
\left(\Gamma_{n+1}^{p}\right)_{(n+1)}&-\Gamma_{n}^{p+1}&\Gamma_{n}^{p}+\Gamma_{n}^{p+2}\end{array}\right|$
$\displaystyle=$
$\displaystyle\left|\Gamma_{n}^{p}+\Gamma_{n}^{p+2}\right|\cdot\left|\begin{array}[]{cc}-\Gamma_{n+1}^{p}&\left(\Gamma_{n+1}^{p+1}\right)^{(n+1)}\\\
\left(\Gamma_{n+1}^{p+1}\right)_{(n+1)}&\Gamma_{n}^{p}\end{array}\right|$
$\displaystyle=$
$\displaystyle\left|\Gamma_{n}^{p}+\Gamma_{n}^{p+2}\right|\cdot\left|\begin{array}[]{cc}-\Gamma_{n+1}^{p}&\boldsymbol{0}_{(n+1)\times
n}\\\
\left(\Gamma_{n+1}^{p+1}\right)_{(n+1)}&\Gamma_{n}^{p}+\Gamma_{n}^{p+2}\end{array}\right|.$

Hence,

$\left|\Delta_{3n+1}^{3p}\right|\equiv(-1)^{n+1}\left|\Gamma_{n+1}^{p}\right|\cdot\left|\Delta_{n}^{p}\right|^{2}.$

$12)$ Combine (2.4) and (2.7), we have

$\displaystyle\left|P^{t}\Delta_{3n+2}^{3p}P\right|$ $\displaystyle\equiv$
$\displaystyle\left|\begin{array}[]{ccc}-\Gamma_{n+1}^{p}&\Gamma_{n+1}^{p+1}&\left(\Gamma_{n+1}^{p}\right)^{(n+1)}\\\
\Gamma_{n+1}^{p+1}&\Gamma_{n+1}^{p}&-\left(\Gamma_{n+1}^{p+1}\right)^{(n+1)}\\\
\left(\Gamma_{n+1}^{p}\right)_{(n+1)}&-\left(\Gamma_{n+1}^{p+1}\right)_{(n+1)}&\Gamma_{n}^{p+2}\end{array}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccc}-\Gamma_{n+1}^{p}&\Gamma_{n+1}^{p+1}&\boldsymbol{0}_{(n+1)\times
n}\\\ \Gamma_{n+1}^{p+1}&\Gamma_{n+1}^{p}&\boldsymbol{0}_{(n+1)\times n}\\\
\left(\Gamma_{n+1}^{p}\right)_{(n+1)}&-\left(\Gamma_{n+1}^{p+1}\right)_{(n+1)}&\Gamma_{n}^{p}+\Gamma_{n}^{p+2}\end{array}\right|.$

By Lemma 1,

$\displaystyle\left|\Delta_{3n+2}^{3p}\right|$ $\displaystyle\equiv$
$\displaystyle\left|\Delta_{n}^{p}\right|\cdot\left|\begin{array}[]{cc}-\Gamma_{n+1}^{p}&\Gamma_{n+1}^{p+1}\\\
\Gamma_{n+1}^{p+1}&\Gamma_{n+1}^{p}\end{array}\right|$ $\displaystyle=$
$\displaystyle(-1)^{n}\left|\Gamma_{n+2}^{p}\right|\cdot\left|\Delta_{n}^{p}\right|^{2}+(-1)^{n+1}\left|\Gamma_{n+1}^{p}\right|\cdot\left|\Delta_{n}^{p}\right|\cdot\left|\Delta_{n+1}^{p}\right|.$

$13)$ Combine (2.4) and (2.5), we have

$\displaystyle\left|P^{t}\Delta_{3n}^{3p+1}P\right|$ $\displaystyle\equiv$
$\displaystyle\left|\begin{matrix}\Gamma_{n}^{p+1}&\Gamma_{n}^{p}&\mathbf{-}\Gamma_{n}^{p+1}\\\
\Gamma_{n}^{p}&\mathbf{-}\Gamma_{n}^{p+1}&\Gamma_{n}^{p+2}\\\
\mathbf{-}\Gamma_{n}^{p+1}&\Gamma_{n}^{p+2}&\Gamma_{n}^{p+1}\end{matrix}\right|$
$\displaystyle=$ $\displaystyle\left|\begin{matrix}\mathbf{0}_{n\times
n}&\Gamma_{n}^{p}+\Gamma_{n}^{p+2}&\mathbf{0}_{n\times n}\\\
\Gamma_{n}^{p}&\mathbf{-}\Gamma_{n}^{p+1}&\Gamma_{n}^{p+2}\\\
\mathbf{-}\Gamma_{n}^{p+1}&\Gamma_{n}^{p+2}&\Gamma_{n}^{p+1}\end{matrix}\right|$
$\displaystyle=$
$\displaystyle(-1)^{n}\left|\Gamma_{n}^{p}+\Gamma_{n}^{p+2}\right|\cdot\left|\begin{matrix}\Gamma_{n}^{p}&\Gamma_{n}^{p+2}\\\
\mathbf{-}\Gamma_{n}^{p+1}&\Gamma_{n}^{p+1}\end{matrix}\right|$
$\displaystyle=$
$\displaystyle(-1)^{n}\left|\Gamma_{n}^{p}+\Gamma_{n}^{p+2}\right|\cdot\left|\begin{matrix}\Gamma_{n}^{p}+\Gamma_{n}^{p+2}&\Gamma_{n}^{p+2}\\\
\mathbf{0}_{n\times n}&\Gamma_{n}^{p+1}\end{matrix}\right|$ $\displaystyle=$
$\displaystyle(-1)^{n}\left|\Gamma_{n}^{p}+\Gamma_{n}^{p+2}\right|^{2}\cdot\left|\Gamma_{n}^{p+1}\right|.$

Therefore,

$\left|\Delta_{3n}^{3p+1}\right|\equiv(-1)^{n}\left|\Gamma_{n}^{p+1}\right|\cdot\left|\Delta_{n}^{p}\right|^{2}.$

$14)$ Combine (2.4) and (2.6), we have

$\displaystyle\left|P^{t}\Delta_{3n+1}^{3p+1}P\right|$ $\displaystyle\equiv$
$\displaystyle\left|\begin{array}[]{ccc}\Gamma_{n+1}^{p+1}&\left(\Gamma_{n+1}^{p}\right)^{(n+1)}&-\left(\Gamma_{n+1}^{p+1}\right)^{(n+1)}\\\
\left(\Gamma_{n+1}^{p}\right)_{(n+1)}&-\Gamma_{n}^{p+1}&\Gamma_{n}^{p+2}\\\
-\left(\Gamma_{n+1}^{p+1}\right)_{(n+1)}&\Gamma_{n}^{p+2}&\Gamma_{n}^{p+1}\end{array}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccc}\Gamma_{n+1}^{p+1}&\left(\Gamma_{n+1}^{p}\right)^{(n+1)}&\mathbf{0}_{(n+1)\times
n}\\\
\left(\Gamma_{n+1}^{p}\right)_{(n+1)}&-\Gamma_{n}^{p+1}&\Gamma_{n}^{p}+\Gamma_{n}^{p+2}\\\
-\left(\Gamma_{n+1}^{p+1}\right)_{(n+1)}&\Gamma_{n}^{p+2}&\mathbf{0}_{n\times
n}\end{array}\right|$ $\displaystyle=$
$\displaystyle(-1)^{n}\left|\Gamma_{n}^{p}+\Gamma_{n}^{p+2}\right|\cdot\left|\begin{array}[]{cc}\Gamma_{n+1}^{p+1}&\left(\Gamma_{n+1}^{p}\right)^{(n+1)}\\\
-\left(\Gamma_{n+1}^{p+1}\right)_{(n+1)}&\Gamma_{n}^{p+2}\end{array}\right|$
$\displaystyle=$
$\displaystyle(-1)^{n}\left|\Gamma_{n}^{p}+\Gamma_{n}^{p+2}\right|\cdot\left|\begin{array}[]{cc}\Gamma_{n+1}^{p+1}&\left(\Gamma_{n+1}^{p}\right)^{(n+1)}\\\
0_{n\times(n+1)}&\Gamma_{n}^{p}+\Gamma_{n}^{p+2}\end{array}\right|.$

Hence,

$\left|\Delta_{3n+1}^{3p+1}\right|\equiv(-1)^{n}\left|\Gamma_{n+1}^{p+1}\right|\cdot\left|\Delta_{n}^{p}\right|^{2}.$

$15)$ Combine (2.4) and (2.7), we have

$\displaystyle\left|P^{t}\Delta_{3n+2}^{3p+1}P\right|$ $\displaystyle\equiv$
$\displaystyle\left|\begin{array}[]{ccc}\Gamma_{n+1}^{p+1}&\Gamma_{n+1}^{p}&-\left(\Gamma_{n+1}^{p+1}\right)^{(n+1)}\\\
\Gamma_{n+1}^{p}&-\Gamma_{n+1}^{p+1}&\left(\Gamma_{n+1}^{p+2}\right)^{(n+1)}\\\
-\left(\Gamma_{n+1}^{p+1}\right)_{(n+1)}&\left(\Gamma_{n+1}^{p+2}\right)_{(n+1)}&\Gamma_{n}^{p+1}\end{array}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccc}\Gamma_{n+1}^{p+1}&\Gamma_{n+1}^{p}&\mathbf{0}_{(n+1)\times
n}\\\ \Gamma_{n+1}^{p}&-\Gamma_{n+1}^{p+1}&\mathbf{0}_{(n+1)\times n}\\\
-\left(\Gamma_{n+1}^{p+1}\right)_{(n+1)}&\left(\Gamma_{n+1}^{p+2}\right)_{(n+1)}&\Gamma_{n}^{p+1}+\Gamma_{n}^{p+3}\end{array}\right|$
$\displaystyle=$
$\displaystyle\left|\Gamma_{n}^{p+1}+\Gamma_{n}^{p+3}\right|\cdot\left|\begin{array}[]{cc}\Gamma_{n+1}^{p+1}&\Gamma_{n+1}^{p}\\\
\Gamma_{n+1}^{p}&-\Gamma_{n+1}^{p+1}\end{array}\right|.$

By Lemma 1,

$\left|\Delta_{3n+2}^{3p+1}\right|\equiv(-1)^{n}\left|\Gamma_{n+2}^{p}\right|\cdot\left|\Delta_{n}^{p}\right|\cdot\left|\Delta_{n}^{p+1}\right|+(-1)^{n+1}\left|\Gamma_{n+1}^{p}\right|\cdot\left|\Delta_{n}^{p+1}\right|\cdot\left|\Delta_{n+1}^{p}\right|.$

$16)$ Combine (2.4) and (2.5), we have

$\displaystyle\left|P^{t}\Delta_{3n}^{3p+2}P\right|$ $\displaystyle\equiv$
$\displaystyle\left|\begin{matrix}\Gamma_{n}^{p}&-\Gamma_{n}^{p+1}&\Gamma_{n}^{p+2}\\\
-\Gamma_{n}^{p+1}&\Gamma_{n}^{p+2}&\Gamma_{n}^{p+1}\\\
\Gamma_{n}^{p+2}&\Gamma_{n}^{p+1}&-\Gamma_{n}^{p+2}\end{matrix}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{matrix}\Gamma_{n}^{p}+\Gamma_{n}^{p+2}&\mathbf{0}_{n\times
n}&\mathbf{0}_{n\times n}\\\
-\Gamma_{n}^{p+1}&\Gamma_{n}^{p+2}&\Gamma_{n}^{p+1}\\\
\Gamma_{n}^{p+2}&\Gamma_{n}^{p+1}&-\Gamma_{n}^{p+2}\end{matrix}\right|$
$\displaystyle=$
$\displaystyle\left|\Gamma_{n}^{p}+\Gamma_{n}^{p+2}\right|\cdot\left|\begin{matrix}\Gamma_{n}^{p+2}&\Gamma_{n}^{p+1}\\\
\Gamma_{n}^{p+1}&-\Gamma_{n}^{p+2}\end{matrix}\right|.$

By Lemma 1,

$\left|\Delta_{3n}^{3p+2}\right|\equiv(-1)^{n}\left|\Gamma_{n}^{p+1}\right|\cdot\left|\Delta_{n}^{p}\right|\cdot\left|\Delta_{n}^{p+1}\right|+(-1)^{n+1}\left|\Gamma_{n+1}^{p+1}\right|\cdot\left|\Delta_{n}^{p}\right|\cdot\left|\Delta_{n-1}^{p+1}\right|.$

$17)$ Combine (2.4) and (2.6), we have

$\displaystyle\left|P^{t}\Delta_{3n+1}^{3p+2}P\right|$ $\displaystyle\equiv$
$\displaystyle\left|\begin{matrix}\Gamma_{n+1}^{p}&-\left(\Gamma_{n+1}^{p+1}\right)^{(n+1)}&\left(\Gamma_{n+1}^{p+2}\right)^{(n+1)}\\\
-\left(\Gamma_{n+1}^{p+1}\right)_{(n+1)}&\Gamma_{n}^{p+2}&\Gamma_{n}^{p+1}\\\
\left(\Gamma_{n+1}^{p+2}\right)_{(n+1)}&\Gamma_{n}^{p+1}&-\Gamma_{n}^{p+2}\end{matrix}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{matrix}\Gamma_{n+1}^{p}&\mathbf{0}_{(n+1)\times
n}&\left(\Gamma_{n+1}^{p+2}\right)^{(n+1)}\\\
-\left(\Gamma_{n+1}^{p+1}\right)_{(n+1)}&\mathbf{0}_{n\times
n}&\Gamma_{n}^{p+1}\\\
\left(\Gamma_{n+1}^{p+2}\right)_{(n+1)}&\Gamma_{n}^{p+1}+\Gamma_{n}^{p+3}&-\Gamma_{n}^{p+2}\end{matrix}\right|$
$\displaystyle=$
$\displaystyle(-1)^{n}\left|\Gamma_{n}^{p+1}+\Gamma_{n}^{p+3}\right|\cdot\left|\begin{matrix}\Gamma_{n+1}^{p}&\left(\Gamma_{n+1}^{p+2}\right)^{(n+1)}\\\
-\left(\Gamma_{n+1}^{p+1}\right)_{(n+1)}&\Gamma_{n}^{p+1}\end{matrix}\right|$
$\displaystyle=$
$\displaystyle(-1)^{n}\left|\Gamma_{n}^{p+1}+\Gamma_{n}^{p+3}\right|\cdot\left|\begin{matrix}\Gamma_{n+1}^{p}&\left(\Gamma_{n+1}^{p+2}\right)^{(n+1)}\\\
0_{n\times(n+1)}&\Gamma_{n}^{p+1}+\Gamma_{n}^{p+3}\end{matrix}\right|.$

Hence,

$\left|\Delta_{3n+1}^{3p+2}\right|\equiv(-1)^{n}\left|\Gamma_{n+1}^{p}\right|\cdot\left|\Delta_{n}^{p+1}\right|^{2}.$

$18)$ Combine (2.4) and (2.7), we have

$\displaystyle\left|P^{t}\Delta_{3n+2}^{3p+2}P\right|$ $\displaystyle\equiv$
$\displaystyle\left|\begin{matrix}\Gamma_{n+1}^{p}&-\Gamma_{n+1}^{p+1}&\left(\Gamma_{n+1}^{p+2}\right)^{(n+1)}\\\
-\Gamma_{n+1}^{p+1}&\Gamma_{n+1}^{p+2}&\left(\Gamma_{n+1}^{p+1}\right)^{(n+1)}\\\
\left(\Gamma_{n+1}^{p+2}\right)_{(n+1)}&\left(\Gamma_{n+1}^{p+1}\right)_{(n+1)}&-\Gamma_{n}^{p+2}\end{matrix}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{matrix}\Gamma_{n+1}^{p}&-\Gamma_{n+1}^{p+1}&\mathbf{0}_{(n+1)\times
n}\\\
-\Gamma_{n+1}^{p+1}&\Gamma_{n+1}^{p+2}&\left(\Gamma_{n+1}^{p+1}+\Gamma_{n+1}^{p+3}\right)^{(n+1)}\\\
\left(\Gamma_{n+1}^{p+2}\right)_{(n+1)}&\left(\Gamma_{n+1}^{p+1}\right)_{(n+1)}&\mathbf{0}_{n\times
n}\end{matrix}\right|$ $\displaystyle=$
$\displaystyle\left|\begin{matrix}\Gamma_{n+1}^{p}&-\Gamma_{n+1}^{p+1}&\mathbf{0}_{(n+1)\times
n}\\\ \begin{array}[]{c}\mathbf{0}_{n\times(n+1)}\\\
-\left(\alpha_{p+n+1}^{n+1}\right)^{t}\end{array}&\begin{array}[]{c}\mathbf{0}_{n\times(n+1)}\\\
\left(\alpha_{p+n+2}^{n+1}\right)^{t}\end{array}&\left(\Gamma_{n+1}^{p+1}+\Gamma_{n+1}^{p+3}\right)^{(n+1)}\\\
\left(\Gamma_{n+1}^{p+2}\right)_{(n+1)}&\left(\Gamma_{n+1}^{p+1}\right)_{(n+1)}&\mathbf{0}_{n\times
n}\end{matrix}\right|$ $\displaystyle=$
$\displaystyle\left|\Gamma_{n}^{p+1}+\Gamma_{n}^{p+3}\right|\cdot\left|\begin{matrix}\Gamma_{n+1}^{p}&-\Gamma_{n+1}^{p+1}\\\
-\left(\alpha_{p+n+1}^{n+1}\right)^{t}&\left(\alpha_{p+n+2}^{n+1}\right)^{t}\\\
\left(\Gamma_{n+1}^{p+2}\right)_{(n+1)}&\left(\Gamma_{n+1}^{p+1}\right)_{(n+1)}\end{matrix}\right|$
$\displaystyle=$
$\displaystyle\left|\Gamma_{n}^{p+1}+\Gamma_{n}^{p+3}\right|\cdot(-1)\left|\begin{matrix}\left(\Gamma_{n+2}^{p}\right)^{(n+2)}&-\left(\Gamma_{n+2}^{p+1}\right)^{(n+2)}\\\
\left(\Gamma_{n+1}^{p+2}\right)_{(n+1)}&\left(\Gamma_{n+1}^{p+1}\right)_{(n+1)}\end{matrix}\right|.$

Note that

$\displaystyle\left|\begin{matrix}\left(\Gamma_{n+2}^{p}\right)^{(n+2)}&-\left(\Gamma_{n+2}^{p+1}\right)^{(n+2)}\\\
\left(\Gamma_{n+1}^{p+2}\right)_{(n+1)}&\left(\Gamma_{n+1}^{p+1}\right)_{(n+1)}\end{matrix}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccc}\left(\Gamma_{n+2}^{p}\right)^{(n+2)}&\mathbf{0}_{(n+2)\times
n}&-\alpha_{p+n+1}^{n+2}\\\
\left(\Gamma_{n+1}^{p+2}\right)_{(n+1)}&\Gamma_{n}^{p+1}+\Gamma_{n}^{p+3}&\alpha_{p+n+1}^{n}\end{array}\right|$
$\displaystyle=$
$\displaystyle(-1)^{n}\left|\Gamma_{n}^{p+1}+\Gamma_{n}^{p+3}\right|\cdot(-1)\left|\Gamma_{n+2}^{p}\right|,$

we have

$\left|\Delta_{3n+2}^{3p+2}\right|\equiv(-1)^{n}\left|\Gamma_{n+2}^{p}\right|\cdot\left|\Delta_{n}^{p+1}\right|^{2}.$

∎