# Exploring Enabling Technologies for Programmable and Software-Defined
Networks: Challenges and Opportunities

David Carrascal  Departamento de Automática
Universidad de Alcalá
Alcala de Henares (Madrid), Spain
<EMAIL_ADDRESS>Elisa Rojas  Departamento de Automática
Universidad de Alcalá
Alcala de Henares (Madrid), Spain
<EMAIL_ADDRESS>Diego Lopez-Pajares  Departamento de Automática
Universidad de Alcalá
Alcala de Henares (Madrid), Spain
<EMAIL_ADDRESS>
# Enabling Technologies for Programmable and Software-Defined Networks:
Challenges, Opportunities, and the Path Towards 6G

David Carrascal  Departamento de Automática
Universidad de Alcalá
Alcala de Henares (Madrid), Spain
<EMAIL_ADDRESS>Elisa Rojas  Departamento de Automática
Universidad de Alcalá
Alcala de Henares (Madrid), Spain
<EMAIL_ADDRESS>Diego Lopez-Pajares  Departamento de Automática
Universidad de Alcalá
Alcala de Henares (Madrid), Spain
<EMAIL_ADDRESS>
# Enabling Technologies for Programmable and Software-Defined Networks:
Bolstering the Path Towards 6G

David Carrascal  Departamento de Automática
Universidad de Alcalá
Alcala de Henares (Madrid), Spain
<EMAIL_ADDRESS>Elisa Rojas  Departamento de Automática
Universidad de Alcalá
Alcala de Henares (Madrid), Spain
<EMAIL_ADDRESS>Diego Lopez-Pajares  Departamento de Automática
Universidad de Alcalá
Alcala de Henares (Madrid), Spain
<EMAIL_ADDRESS>
###### Abstract

Although the complete scope of the sixth generation of mobile technologies
(6G) is still unclear, the prominence of the Internet of Things (IoT) and
Artificial Intelligence (AI) / Machine Learning (ML) in the networking field
is undeniable. In this regard, key technology enablers for the previous
generation, 5G, such as software-defined networking and network function
virtualization, fall short to accomplish the stringent requirements envisioned
for 6G verticals. This PhD thesis goes back to basics, by exploring missing
functionality gaps in relation to these technologies, in order to provide the
“glue” for holistic and fully-fledged networking solutions for 6G, aligned
with standards and industry recommendations. Although ambitious, and in a very
early stage, this PhD thesis illustrates an initial design for in-band control
in Software-Defined Networking (SDN) that could facilitate the
interoperability among constrained IoT devices. The current design
demonstrates promising results in terms of resource-usage and robustness,
which are pivotal features for constrained networks. Next steps include the
integration of the approach with a real testbed comprised of constrained IoT
devices and the implementation of a federated learning environment at the
edge.

###### Index Terms:

SDN, IoT, In-band Control, Programmable Data Planes, Edge Computing, 5G/6G

##  I Introduction

Since the successful launch of the first commercial deployments of the fifth
generation of mobile technologies (5G), there is growing interest in the
research and innovation results and applications that can be achieved with
both 5G and beyond (i.e. the sixth generation of mobile technologies (6G)) [1,
2]. One cornerstone of 5G networks is founded on moving part of the network
intelligence from the core of the network to the edge of the network (users
and/or things), enabling what is known as mist/fog/edge computing, or edge-to-
cloud continuum [3], and providing a multitude of advantages and fostering
novel applications. It is expected that 6G networks will further build on this
paradigm, thanks to the increase in internet-enabled devices [2], and at the
same time, this paradigm is strongly supported by Software-Defined Networking
(SDN) and Network Function Virtualization (NFV) as key technologies in its
infrastructure [4] [5]. Both SDN and NFV technologies are therefore radically
transforming conventional communications networks, as well as posing new
challenges and strategies in the upcoming 6G commercial communications
networks [6]. However, achieving this new computational model and
communication infrastructure still involves overcoming several challenges
associated with the foundational components of both the data plane and the
control plane of SDN/NFV-based communication networks. For instance, few
research efforts consider the potential constraints of memory and energy of
Internet of Things (IoT) devices for their integration into these softwarized
environments. Consequently, there is a niche for improvement and optimisation
in these technologies, particularly for distributed and constrained
environments as those envisioned by the edge-to-cloud continuum.

This PhD thesis investigates into fundamental technologies for the
implementation of 6G verticals, with a particular emphasis on constrained IoT
devices at the edge and, thus, functional features such as scalability,
robustness, and flexibility, are pivotal in the design process. In the next
sections, we describe the state of the art, objectives and research
methodology, some preliminary research ideas, and finally we conclude the
paper by providing insights and next steps.

##  II State of the art

Figure 1: Test architecture design pipeline for the research methodology

As a further leap from 5G, 6G [7] is shaping up to be a revolution in mobile
connectivity, with even faster speed, lower latency and greater data capacity.
But it is not just about speed and capacity: 6G is also focused on creating
smart, sustainable networks to facilitate the green transition, which means a
major impact on several key sectors of the economy. 6G verticals include
energy, transportation, industry, healthcare, cities and utilities, media and
entertainment, tourism and culture [8]. Each of these verticals have specific
needs and demands in terms of connectivity and sustainability, and 6G promises
to address these needs and move towards a smarter, more connected society. For
instance, one use case envisions the creation of smart grids for the green
transition, which implies greater energy efficiency and the reduction of
greenhouse gas emissions in the energy sector [9]. Moreover, 6G could improve
connectivity in the transportation sector, enabling autonomous vehicles and
improved road safety. It could also have a significant impact on healthcare,
with the creation of more advanced telemedicine networks and improved remote
care. In the media and entertainment sector, 6G could improve the quality of
video and audio transmission, as well as enable the creation of new augmented
and virtual reality experiences.

In the context of 6G, SDN is a key enabler for the creation of smart and
sustainable networks, as it enhances management of network resources (e.g.,
implementation of green transition policies in real time [10]), and it adjusts
to the needs of different 6G verticals, facilitating the design of
personalized services and improving the user experience.

Nevertheless, although SDN is already a mature technology, it still suffers
from diverse limitations, including security aspects [11]. For example, 6G
envisions multiple type of connectivity links for diverse types of IoT devices
[12, 13], even working together and located in different areas (earth, heaven
and sea) [14], but IoT devices at the edge are usually constrained in memory
and energy. As such, they cannot support a standard out-of-band SDN channel;
while using in-band control could be recommended, its deployment is still
manually performed in most scenarios since automatization has been barely
addressed in literature [15]. Associated verticals that would benefit from
scalable and automatic in-band control deployment include sparsely distributed
federated learning environments for smart agriculture or local redistribution
of computing capabilities for mobile tactical edge environments [16], for
example. In summary, among other improvements of the SDN paradigm, the in-band
paradigm is a promising approach for IoT network management in 6G
environments, and its proper implementation could enable more efficient and
scalable management of IoT devices in the aforementioned different verticals
and use cases [17, 18].

##  III Objectives & research methodology

According to the previous analysis of the state of the art, which is outlined
as a mere starting point, the following research questions are contemplated:

  * •

What enabling technologies are missing in the SDN/NFV puzzle to be applicable
to all type of heterogeneous networks?

  * •

How to integrate Artificial Intelligence (AI)/ Machine Learning (ML) into SDN
to enhance network behavior and, vice versa, how to leverage SDN for advanced
AI/ML services?

  * •

How can security and privacy be ensured in a 6G network composed of IoT
devices?

  * •

Is it possible to involve, and share, the resources of end-user devices (with
the privacy issues that this raises) such as mobile, Augmented Reality (AR)
and Virtual Reality (VR) devices in the network edge?

  * •

To what extent novel structures generated at the data plane could increase
network programmability and bolster SDN/NFV?

The research methodology has been adapted and modified according to the
specific needs of the research to be carried out. This methodology, depicted
in Fig. 1, is mainly based on the wide experience of the research group and
can be divided into the following steps. Firstly, a theoretical study of the
state of the art and found solutions, analysing their feasibility and
performance, in order to take advantage as much as possible of existing works.
Our intention is to prove which works are long-term and foundational ideas, to
build upon them, and which are not, to look for potential conceptual gaps.
This thesis is located at a very low Technology Readiness Level (TRL) and this
initial step is essential. Afterwards, we design and implement our ideas in
specific test platforms. We like to acknowledge results both via simulation
and emulation. Finally, a test pipeline will be developed to stress the
implemented solutions in order to evaluate their behavior under real
conditions, their feasibility, scalability and performance. Figure 1
summarizes this process, listing some of the most common platforms we use and
how relying on a random topology generator is pivotal for comprehensive
testbenching in early research stages.

##  IV Preliminary research ideas

The very first research ideas of the author’s PhD thesis are focused to design
and scalable and resilient in-band SDN control protocol for constrained mobile
devices at the network edge. After a comprehensive survey of the state of the
art [15] and an initial prototype for in-band in wired/wireless hybrid SDN
scenarios implemented by the PhD student’s research group [19], the objective
is to re-design the approach reducing memory and energy usage as much as
possible, while building robust paths and keeping in-band functionality too.

Figure 2: Hierarchical labelling for in-band control in IoT networks

Since the PhD student had previous knowledge about programmability of IoT data
planes [20] and about IoT routing protocols [21], merging both approaches had
led to the design of the labelling protocol depicted in Fig. 2. The idea is as
follows: Let us consider a set of IoT devices, as in Fig. 2, one (or more)
will behave as gateways (that is, they will be directly connected, or have an
already existing path, to an SDN controller). At the same time, all nodes
might broadcast a Hello-like message, so that neighbors are locally aware of
what other IoT devices can be reached. With that list of neighbors, gateways
will self-determine themselves as such and emit a labelling probe frame to
each neighbor, which is received and re-sent with a label update to other
neighbors, hence, exploring the whole network.

In the case of Fig. 2, the gateway has label 1, and sends label 1.2 to its
only neighbor, by simplifying appending the neighbor ID (2), which is known
from the neighbor list, to its own label. Afterwards, the node in the middle
forwards labels 1.2.3 and 1.2.4, and also a label 1.2.X back to the gateway.
However, the gateway discards such a label following the criterion that its
own one 1 is comprised, as a prefix, in 1.2.X (independently of the value of
X). This latter mechanism grants no forwarding loops are created in the
network. Finally, nodes 1.2.3 and 1.2.4 will also send labels with their
neighbors, including 1.2, which will discard any label from them to avoid
loops, once again. More specifically, 1.2.3 receives 1.2.4.3, and 1.2.4
obtains 1.2.3.4. The labelling process finalizes when no additional message is
exchanged, after the remaining ones are discarded. Additionally, the mechanism
is compatible with more than one network gateway, even if they assign
themselves the same initial label, though this slightly reduces the total
amount of assigned labels.

Once labels are set, the in-band routing is based on the principles of the
revision of the IEEE standard 802-2001 published in 2014 [22], that is, nodes
can leverage those labels as their own pseudo-MAC addresses and source-route
frames until reaching the gateway. For example, node 1.2.3/1.2.4.3 has two
possible routes towards the gateway, one for 1.2.3, traversing three nodes,
and another for 1.2.4.3, through four. Selecting one path or another could be
based on diverse metrics: fewer hops, smallest latency, etc.

The main benefits of this approach are:

  1. 1.

Nodes have as many routes to the gateway as assigned labels (which grants
smooth recovery and almost instant route modifications in case of link failure
or mobility).

  2. 2.

Table size does not depend on the network size, but only on the amount of
neighbors and amount of assigned labels (for instance, four entries for node
1.2.3, as it has two neighbors and two labels, independently of the rest of
the network).

  3. 3.

The whole label assignment process simply requires two messages per node: one
Hello message and another message to assign labels to neighbors. Of course, in
case of high mobility, the number of messages should be tuned.

Additionally, simulations of the labelling process with Contiki-ng and Cooja
demonstrate very low converge times for the whole assignment procedure (of
around 1 second for 200-node networks [21]).

On the other hand, this idea also has limitations about the label length
(limited to the length of the MAC address field if the frame is not modified)
and potential security issues (concerning the use of Hellos and probe frames).

To acknowledge the feasibility of the whole in-band protocol, the PhD student
is currently working on implementing the design in the specific environment of
SDN, using the BOFUSS software switch [23], as it easier to modify than Open
vSwitch (OVS). The BOFUSS switch is currently being conceptually tested and
ported to the latest versions of Linux, to be integrated with Mininet-WiFi.

Nevertheless, this is just a departing point. Afterwards, diverse ideas,
explored in parallel, are envisioned as future steps, including: enhancing the
security of the in-band protocol, leveraging the protocol for sharing of
computational resources in a distributed manner (for example, to implement
federated learning approaches in AR/VR environments), or integrating the idea
into a more holistic project (for instance, in the field of ETSI standards for
5G and beyond: NFV, Multi-access Edge Computing (MEC), and the current
TeraFlowSDN controller [24]).

##  V Conclusion and next steps

This early-stage PhD thesis investigates key technology enablers to bolster
the verticals envisioned by 6G and, more specifically, in fulfilling the
missing functionality gaps for SDN and programmable networks. So far, we have
presented the design of a scalable and resilient in-band control approach for
IoT, which will be implemented in a Raspberry Pi-based environment based on
BOFUSS. Preliminary tests show that the algorithm is capable of creating
multiple dynamic paths per node towards the gateway in around 1 second for
200-node networks, which facilitates smooth updates for mobility and link-
failure scenarios, and requires small tables (independently of network size)
and few exchanged control messages, which is particularly desirable for
constrained IoT environments.

Regarding future research steps, we will maintain the focus on novel
communication protocols and algorithms that enable the programmability of the
control and data planes. Additionally, the automation of network management
through artificial intelligence could be explored further. In the long term,
as a final step of the thesis, we envisage a prototype of federated vision
learning, integrated with VR and digital twinning, for a smart agriculture
environment. We are currently discussing potential applications with two
stakeholders from Guadalajara and Madrid, for traditional and innovative
farming, respectively.

## Acknowledgment

This work was funded by grants from Comunidad de Madrid through projects
TAPIR-CM (S2018/TCS-4496) and MistLETOE-CM (CM/JIN/2021-006), by project
ONENESS (PID2020-116361RA-I00) of the Spanish Ministry of Science and
Innovation, and by Universidad de Alcalá through “Contratos Predoctorales de
Formación de Personal Investigador - FPI-UAH 2022”.

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  *[5G]: the fifth generation of mobile technologies
  *[6G]: the sixth generation of mobile technologies
  *[SDN]: Software-Defined Networking
  *[NFV]: Network Function Virtualization
  *[IoT]: Internet of Things
  *[AI]: Artificial Intelligence
  *[ML]: Machine Learning
  *[AR]: Augmented Reality
  *[VR]: Virtual Reality
  *[TRL]: Technology Readiness Level
  *[OVS]: Open vSwitch
  *[MEC]: Multi-access Edge Computing


# Factoring Variants of Chebyshev Polynomials
with Minimal Polynomials of $\cos(\frac{2\pi}{d})$

D.A. Wolfram  College of Engineering & Computer Science
The Australian National University, Canberra, ACT 0200
<EMAIL_ADDRESS>
###### Abstract

We solve the problem of factoring polynomials $V_{n}(x)\pm 1$ and $W_{n}(x)\pm
1$ where $V_{n}(x)$ and $W_{n}(x)$ are Chebyshev polynomials of the third and
fourth kinds. The method of proof is based on previous work by Wolfram [12]
for factoring variants of Chebyshev polynomials of the first and second kinds,
$T_{n}(x)\pm 1$ and $U_{n}(x)\pm 1$. We also show that, in general, there are
no factorizations of variants of Chebyshev polynomials of the fifth and sixth
kinds, $X_{n}(x)\pm 1$ and $Y_{n}(x)\pm 1$ using minimal polynomials of
$\cos(\frac{2\pi}{d})$.

00footnotetext: MSC: Primary 12E10, Secondary 12D05

##  1 Background.

Chebyshev polynomials of the third and fourth kinds were named by Gautschi [7]
and are also called airfoil polynomials [6, 11]. They are used in areas such
as solving differential equations [1], numerical integration [4, 6, 11],
approximations [11], interpolation [7] and combinatorics [5]. The significance
of their applications in mathematics, engineering and numerical modeling
provides a motivation for studying the properties of these polynomials.

In previous work, Wolfram [12] solved an open factorization problem for
Chebyshev polynomials of the second kind $U_{n}(x)\pm 1$, and gave a more
direct proof of the result for Chebyshev polynomials of the first kind,
$T_{n}(x)\pm 1$ . We apply this method to solve the analogous factorization
problems for Chebyshev polynomials of the third and fourth kinds. All of these
factorizations can be expressed in terms of the minimal polynomials of
$\cos(\frac{2\pi}{d})$.

###  1.1 Chebyshev Polynomials of the Second Kind

Chebyshev polynomials of the second kind can be defined by

$U_{n}(x)=\frac{\sin((n+1)\theta)}{\sin(\theta)}$ (1)

where $x=\cos\theta$ and $n\geq 0$. This is equation (1.4) of Mason and
Handscomb [11]. It follows that

$U_{n}(x)^{2}-1=U_{n-1}(x)U_{n+1}(x)$ (2)

where $n\geq 1$, by applying the trigonometric identity
$\sin^{2}A-\sin^{2}B=\sin(A+B)\sin(A-B)$ with $A=(n+1)\theta$ and $B=\theta$.

These polynomials satisfy the following recurrence, for example equations
(1.6a)–(1.6b) of [11]:

$\displaystyle U_{0}(x)$ $\displaystyle=1,$ $\displaystyle U_{1}(x)$
$\displaystyle=2x,$ $\displaystyle U_{n}(x)=2xU_{n-1}(x)-U_{n-2}(x)$ (3)

where $n>1$.

Gürtaş [8] showed that

$U_{n-1}(x)=\prod_{\mathclap{\begin{subarray}{c}d\mid 2n\\\
d>2\end{subarray}}}\Psi_{d}(x)$ (4)

where $n\geq 1$. The polynomials $\Psi_{d}(x)$ are

$\Psi_{d}(x)=\prod_{\mathclap{\begin{subarray}{c}k\in
S_{d/2}\end{subarray}}}2\left(x-\cos\left(2\pi\frac{k}{d}\right)\right)$ (5)

where $S_{d/2}=\\{k\mid(k,d)=1,1\leq k<d/2\\}$ and $d>2$. They have degree
$\phi(d)/2$ where $\phi$ is Euler’s totient function [8]. The minimal
polynomial in $\mathbb{Q}[x]$ of $\cos(\frac{2\pi}{d})$ is
$2^{-\frac{\phi(d)}{2}}\Psi_{d}(x)$ where $d>2$ which follows from the proof
of Theorem 1 of D. H. Lehmer [9].

######  Definition 1.

The polynomial $\Psi_{1}(x)=2(x-1)$ and $\Psi_{2}(x)=2(x+1)$.

These are polynomials with roots $\cos(2\pi)$ and of $\cos(\pi)$,
respectively.

###  1.2 Chebyshev Polynomials of the Third and Fourth Kinds

Chebyshev polynomials of the third kind can be defined by

$V_{n}(x)=\frac{\cos(n+\frac{1}{2})\theta}{\cos\frac{\theta}{2}}$ (6)

and of the fourth kind by

$W_{n}(x)=\frac{\sin(n+\frac{1}{2})\theta}{\sin\frac{\theta}{2}}$ (7)

where $x=\cos\theta$ and $n\geq 0$.

They can also be defined with respect to Chebyshev polynomials of the second
kind, $U_{n}(x)$:

$V_{n}(x)=U_{n}(x)-U_{n-1}(x)$ (8)

and

$W_{n}(x)=U_{n}(x)+U_{n-1}(x)$ (9)

where $n\geq 1$. These are equations (1.17)–(1.18) of Mason and Handscomb
[11].

##  2 Solution

The method of solution follows that by Wolfram [12].The first step is to
express $V_{n}(x)^{2}-1$ and $W_{n}(x)^{2}-1$ in terms of the minimal
polynomials $\Psi_{d}(x)$ where $d\geq 1$.

######  Lemma 1.

Where $n\geq 1$,

$V_{n}(x)^{2}-1=\Psi_{1}(x)\prod_{\mathclap{\begin{subarray}{c}d\mid 2n\\\
d>2\end{subarray}}}\Psi_{d}(x)\prod_{\mathclap{\begin{subarray}{c}d\mid
2n+2\\\ d>2\end{subarray}}}\Psi_{d}(x)$ (10)
$W_{n}(x)^{2}-1=\Psi_{2}(x)\prod_{\mathclap{\begin{subarray}{c}d\mid 2n\\\
d>2\end{subarray}}}\Psi_{d}(x)\prod_{\mathclap{\begin{subarray}{c}d\mid
2n+2\\\ d>2\end{subarray}}}\Psi_{d}(x).$ (11)

###### Proof.

From equation (8), we have

$\displaystyle V_{n}(x)^{2}-1=$
$\displaystyle(U_{n}(x)-U_{n-1}(x)+1)(U_{n}(x)-U_{n-1}(x)-1)\mbox{ from
equation (\ref{VU})}$ $\displaystyle=$ $\displaystyle
U_{n}(x)^{2}-1-2U_{n}(x)U_{n-1}(x)+U_{n-1}(x)^{2}$ $\displaystyle=$
$\displaystyle U_{n+1}(x)U_{n-1}(x)-2U_{n}(x)U_{n-1}(x)+U_{n-1}(x)^{2}\mbox{
from equation (\ref{UTuran})}$ $\displaystyle=$ $\displaystyle
U_{n-1}(x)(U_{n+1}(x)-2U_{n}(x)+U_{n-1}(x))$ $\displaystyle=$ $\displaystyle
U_{n-1}(x)(2xU_{n}(x)-2U_{n}(x))\mbox{ from equation (\ref{Urec})}$
$\displaystyle=$ $\displaystyle\Psi_{1}(x)U_{n-1}(x)U_{n}(x)\mbox{ from
Definition \ref{def1}}$ $\displaystyle=$
$\displaystyle\Psi_{1}(x)\prod_{\mathclap{\begin{subarray}{c}d\mid 2n\\\
d>2\end{subarray}}}\Psi_{d}(x)\prod_{\mathclap{\begin{subarray}{c}d|2n+2\\\
d>2\end{subarray}}}\Psi_{d}(x)\mbox{ from equation (\ref{UPsi}).}$

Similarly, we have

$\displaystyle W_{n}(x)^{2}-1=$
$\displaystyle(U_{n}(x)+U_{n-1}(x)+1)(U_{n}(x)+U_{n-1}(x)-1)\mbox{ from
equation (\ref{WU})}$ $\displaystyle=$ $\displaystyle
U_{n}(x)^{2}-1+2U_{n}(x)U_{n-1}(x)+U_{n-1}(x)^{2}$ $\displaystyle=$
$\displaystyle U_{n+1}(x)U_{n-1}(x)+2U_{n}(x)U_{n-1}(x)+U_{n-1}(x)^{2}\mbox{
from equation (\ref{UTuran})}$ $\displaystyle=$ $\displaystyle
U_{n-1}(x)(U_{n+1}(x)+2U_{n}(x)+U_{n-1}(x))$ $\displaystyle=$ $\displaystyle
U_{n-1}(x)(2xU_{n}(x)+2U_{n}(x))\mbox{ from equation (\ref{Urec})}$
$\displaystyle=$ $\displaystyle\Psi_{2}(x)U_{n-1}(x)U_{n}(x)\mbox{ from
Definition \ref{def1}}$ $\displaystyle=$
$\displaystyle\Psi_{2}(x)\prod_{\mathclap{\begin{subarray}{c}d\mid 2n\\\
d>2\end{subarray}}}\Psi_{d}(x)\prod_{\mathclap{\begin{subarray}{c}d|2n+2\\\
d>2\end{subarray}}}\Psi_{d}(x)\mbox{ from equation (\ref{UPsi})}$

as required. ∎

The following theorem solves the factorization problem for $V_{n}(x)^{2}-1$.
The second step of the method concerns defining the mapping that splits the
$2n$ factors of $V_{n}(x)^{2}-1$ into the $n$ factors of $V_{n}(x)+1$ and the
other $n$ factors of $V_{n}(x)-1$. The factorizations are unique up to
associativity and commutativity of multiplication.

######  Theorem 1.

If $n\geq 1$,

$V_{n}(x)+1=\prod_{\mathclap{\begin{subarray}{c}d\mid 2n\\\ d>2\\\
2n/d\>\mbox{\rm\small odd}\end{subarray}}}\Psi_{d}(x)\ \ \ \
\prod_{\mathclap{\begin{subarray}{c}d\mid 2n+2\\\ d>2\\\
(2n+2)/d\>\mbox{\rm\small odd}\end{subarray}}}\Psi_{d}(x)$ (12)

and

$V_{n}(x)-1=\Psi_{1}(x)\prod_{\mathclap{\begin{subarray}{c}d\mid 2n\\\ d>2\\\
2n/d\>\mbox{\rm\small even}\end{subarray}}}\Psi_{d}(x)\ \ \ \
\prod_{\mathclap{\begin{subarray}{c}d\mid 2n+2\\\ d>2\\\
(2n+2)/d\>\mbox{\rm\small even}\end{subarray}}}\Psi_{d}(x).$ (13)

###### Proof.

The polynomial $\Psi_{1}(x)=2(x-1)$ is a factor of $V_{n}(x)^{2}-1$ from
equation (10), and $\Psi_{1}(\cos(2\pi))=0$. It follows from equation (6) that
$V_{n}(\cos(2\pi))=1$ and so $\Psi_{1}(x)$ is a factor of $V_{n}(x)-1$.

If $d\mid 2n$ and $d>2$, let $\theta=\frac{2\pi k}{d}$ where $(k,d)=1$, $1\leq
k<\frac{d}{2}$ and $a=\frac{2n}{d}$. We have $\theta=\frac{\pi ak}{n}$ and
$\Psi_{d}(\cos(\theta))=0$. From equation (6),

$\displaystyle V_{n}(\cos(\theta))=$
$\displaystyle\frac{\cos((n+\frac{1}{2})\frac{\pi
ak}{n})}{\cos(\frac{\theta}{2})}$ $\displaystyle=$
$\displaystyle\frac{\cos(\pi ak)\cos(\frac{\theta}{2})-\sin(\pi
ak)\sin(\frac{\theta}{2})}{\cos(\frac{\theta}{2})}.$

The denominator $\cos(\frac{\theta}{2})\not=0$ because
$\frac{\theta}{2}=\frac{\pi k}{d}$ cannot equal $\frac{\pi}{2}$ when $d>2$.
The numbers $ak$ and $a$ have the same parity. This is immediate when $a$ is
even. If $a$ is odd, it follows that $d$ is even and $k$ is odd because
$(k,d)=1$. We have $\cos(\pi ak)=\cos(\pi a)$, and
$V_{n}(\cos(\theta))=\cos(\pi a)$.

Hence, if $a$ is even, then $V_{n}(\cos(\theta))=1$ and $\Psi_{d}(x)$ is a
factor of $V_{n}(x)-1$. Similarly, if $a$ is odd, then
$V_{n}(\cos(\theta))=-1$ and $\Psi_{d}(x)$ is a factor of $V_{n}(x)+1$.

If $d\mid 2n+2$ and $d>2$, let $b=\frac{2n+2}{d}$. We have $\theta=\frac{\pi
bk}{n+1}$ where $k$ is such that $(k,d)=1$ and $1\leq k<\frac{d}{2}$. From
equation (6),

$\displaystyle V_{n}(\cos(\theta))=$
$\displaystyle\frac{\cos((n+\frac{1}{2})\theta)}{\cos(\frac{\theta}{2})}$
$\displaystyle=$
$\displaystyle\frac{\cos((n+1)\theta-\frac{\theta}{2})}{\cos(\frac{\theta}{2})}$
$\displaystyle=$ $\displaystyle\frac{\cos(\pi
bk)\cos(\frac{\theta}{2})+\sin(\pi
bk)\sin(\frac{\theta}{2})}{\cos(\frac{\theta}{2})}.$

Similarly to the previous case the denominator $\cos(\frac{\theta}{2})\not=0$,
the numbers $bk$ and $b$ have the same parity, and
$V_{n}(\cos(\theta))=\cos(\pi b)$. Ift follows that if $b$ is odd then
$\Psi_{d}(x)$ is a factor of $V_{n}(x)+1$ and if $b$ is even then
$\Psi_{d}(x)$ is a factor of $V_{n}(x)-1$.

From equations (3) and (8), $V_{n}(x)$ has degree $n$. It follows that the
right side of equation (10) of the factorization of $V_{n}(x)^{2}-1$ has
degree $2n$. It has $2n$ factors of the form $2(x-\cos(\theta))$ from equation
(5) and Definition 1, half of which are the factors of $V_{n}(x)+1$ and the
other half are the factors of $V_{n}(x)-1$. The mapping defined above maps
every such factor of $V_{n}(x)^{2}-1$ to either $V_{n}(x)+1$ or $V_{n}(x)-1$
depending on whether $\cos(\theta)$ is either a root of $V_{n}(x)+1$ or
$V_{n}(x)-1$, respectively. The right sides of equations (12) and (13) are the
products of these mapped factors and so both have degree equal to $n$.

From equations (3) and (8) and, the leading coefficients of $V_{n}(x)\pm 1$
are $2^{n}$. The expansions of the factorizations on the right sides of
equations (12) and (13) both have $2^{n}$ as leading coefficients also,
because each is a product of $n$ factors of the form $2(x-\cos(\theta))$, as
required. ∎

##  3 Examples with $V$

The polynomial $V_{12}(x)^{2}-1$ has 24 factors, and $V_{12}(x)+1$ and
$V_{12}(x)-1$ each are the products of half of these factors. The mapping in
the proof of Theorem 1 gives

$\displaystyle V_{12}(x)+1=$
$\displaystyle\Psi_{8}(x)\Psi_{24}(x)\Psi_{26}(x)$ $\displaystyle
V_{12}(x)-1=$
$\displaystyle\Psi_{1}(x)\Psi_{3}(x)\Psi_{4}(x)\Psi_{6}(x)\Psi_{12}(x)\Psi_{13}(x)$
$\displaystyle=$ $\displaystyle(2(x-1))(2x+1)(2x)(2x-1)(4x^{2}-3)$
$\displaystyle(64x^{6}+32x^{5}-80x^{4}-32x^{3}+24x^{2}+6x-1)$

The following theorem solves the factorization problem for $W_{n}(x)^{2}-1$.
These factorizations are also unique up to associativity and commutativity of
multiplication.

######  Theorem 2.

If $n\geq 1$,

$W_{n}(x)+1=\prod_{\mathclap{\begin{subarray}{c}d\mid 2n\\\ d>1\\\
2n/d\>\mbox{\rm\small odd}\end{subarray}}}\Psi_{d}(x)\ \ \ \
\prod_{\mathclap{\begin{subarray}{c}d\mid 2n+2\\\ d>2\\\
(2n+2)/d\>\mbox{\rm\small even}\end{subarray}}}\Psi_{d}(x)$ (14)

and

$W_{n}(x)-1=\prod_{\mathclap{\begin{subarray}{c}d\mid 2n\\\ d>1\\\
2n/d\>\mbox{\rm\small even}\end{subarray}}}\Psi_{d}(x)\ \ \ \
\prod_{\mathclap{\begin{subarray}{c}d\mid 2n+2\\\ d>2\\\
(2n+2)/d\>\mbox{\rm\small odd}\end{subarray}}}\Psi_{d}(x).$ (15)

###### Proof.

The structure of the proof is similar to that of Theorem 1. If $d\mid 2n$ and
$d>1$, let $a=\frac{2n}{d}$ and $k$ be such that $(k,d)=1$ where $1\leq
k<\frac{d}{2}$. From equation (7),

$\displaystyle W_{n}(\cos(\theta))=$
$\displaystyle\frac{\sin((n+\frac{1}{2})\frac{\pi
ak}{n})}{\sin(\frac{\theta}{2})}$ $\displaystyle=$
$\displaystyle\frac{\cos(\pi ak)\sin(\frac{\theta}{2})+\sin(\pi
ak)\cos(\frac{\theta}{2})}{\sin(\frac{\theta}{2})}.$

The denominator $\sin(\frac{\theta}{2})\not=0$ because
$\frac{\theta}{2}=\frac{\pi k}{d}$ cannot equal $\pi$ when $d>1$. Similarly,
we have that $ak$ and $a$ have the same parity, and
$W_{n}(\cos(\theta))=\cos(\pi a)$. Hence, if $a$ is even, then
$W_{n}(\cos(\theta))=1$ and $\Psi_{d}(x)$ is a factor of $W_{n}(x)-1$. If $a$
is odd, then $W_{n}(\cos(\theta))=-1$ and $\Psi_{d}(x)$ is a factor of
$W_{n}(x)+1$.

If $d\mid 2n+2$ and $d>2$, let $b=\frac{2n+2}{d}$ and $k$ be such that
$(k,d)=1$ where $1\leq k<\frac{d}{2}$. We have $\theta=\frac{\pi bk}{n+1}$ and
$\frac{\theta}{2}=\frac{\pi k}{d}$. From equation (7),

$\displaystyle W_{n}(\cos(\theta))=$
$\displaystyle\frac{\sin((n+\frac{1}{2})\theta)}{\sin(\frac{\theta}{2})}$
$\displaystyle=$
$\displaystyle\frac{\sin((n+1)\theta-\frac{\theta}{2})}{\sin(\frac{\theta}{2})}$
$\displaystyle=$ $\displaystyle\frac{-\cos(\pi
bk)\sin(\frac{\theta}{2})+\sin(\pi
bk)\cos(\frac{\theta}{2})}{\sin(\frac{\theta}{2})}.$

The denominator $\sin(\frac{\theta}{2})\not=0$, and $b$ and $bk$ have the same
parity, as above. Hence, if $b$ is even, then $W_{n}(\cos(\theta))=-1$ and
$\Psi_{d}(x)$ is a factor of $W_{n}(x)+1$. If $b$ is odd, then
$W_{n}(\cos(\theta))=1$ and $\Psi_{d}(x)$ is a factor of $W_{n}(x)-1$.

It is straightforward to show that the degrees of the right sides of equations
(14) and (15) are both $n$, and the leading coefficients of both sides of
these equations is $2^{n}$. ∎

##  4 Examples with $W$

The polynomial $W_{12}(x)^{2}-1$ has 24 factors, and $W_{12}(x)+1$ and
$W_{12}(x)-1$ each are the products of half of these factors. The mapping in
the proof of Theorem 2 gives

$\displaystyle W_{12}(x)+1=$
$\displaystyle\Psi_{8}(x)\Psi_{24}(x)\Psi_{13}(x)$ $\displaystyle
W_{12}(x)-1=$
$\displaystyle\Psi_{2}(x)\Psi_{3}(x)\Psi_{4}(x)\Psi_{6}(x)\Psi_{12}(x)\Psi_{26}(x)$
$\displaystyle=$ $\displaystyle(2(x+1))(2x+1)(2x)(2x-1)(4x^{2}-3)$
$\displaystyle(64x^{6}-32x^{5}-80x^{4}+32x^{3}+24x^{2}-6x-1)$

When $n$ is odd, $\Psi_{2}(x)$ is a factor of $W_{n}(x)+1$:

$\displaystyle W_{11}(x)+1=$
$\displaystyle\Psi_{2}(x)\Psi_{22}(x)\Psi_{3}(x)\Psi_{4}(x)\Psi_{6}(x)\Psi_{12}(x).$

##  5 Chebyshev Polynomials of the Fifth and Sixth Kinds

Chebyshev polynomials of the fifth kind, $X_{n}(x)$, and sixth kind,
$Y_{n}(x)$, were defined by Masjed-Jamei [10]. Similarly to the other four
kinds of Chebyshev polynomials, they are orthogonal polynomials with integer
coefficients and $X_{n}(x)$ and $Y_{n}(x)$ have degree $n$ where $n\geq 0$ [2,
3]. They have the form

$\sum_{v=0}^{\lfloor\frac{n}{2}\rfloor}a_{v}x^{n-2v}.$

Monic Chebyshev polynomials of the fifth and sixth kinds, $\bar{X}_{n}(x)$ and
$\bar{Y}_{n}(x)$, can be defined by the following recurrences which we
simplify from [2].

$\displaystyle G_{0,m}(x)=$ $\displaystyle 1$ $\displaystyle G_{1,m}(x)=$
$\displaystyle x$ $\displaystyle G_{n,m}(x)=$ $\displaystyle
xG_{n-1,m}(x)+A_{n-1,m}\;G_{n-2,m}(x),\;\;\mbox{ $n>1$}$ (16)

where

$\displaystyle A_{n,m}=$
$\displaystyle\frac{(2n+m-2)(-1)^{n}+(2n-(m-2))-nm-n^{2}}{(2n+m-1)(2n+m-3)},\;\;\mbox{
and}$ (17) $\displaystyle\bar{X}_{n}(x)=$ $\displaystyle G_{n,3}(x)$ (18)
$\displaystyle\bar{Y}_{n}(x)=$ $\displaystyle G_{n,5}(x).$ (19)

The first six Chebyshev polynomials of the fifth kind over $\mathbb{Z}$ are

$\displaystyle X_{0}(x)=$ $\displaystyle 1$ $\displaystyle X_{1}(x)=$
$\displaystyle x$ $\displaystyle X_{2}(x)=$ $\displaystyle 4x^{2}-3$
$\displaystyle X_{3}(x)=$ $\displaystyle 6x^{3}-5x$ $\displaystyle X_{4}(x)=$
$\displaystyle 16x^{4}-20x^{2}+5$ $\displaystyle X_{5}(x)=$ $\displaystyle
80x^{5}-112x^{3}+35x$ $\displaystyle X_{6}(x)=$ $\displaystyle
64x^{6}-112x^{4}+56x^{2}-7.$

The first six Chebyshev polynomials of the sixth kind over $\mathbb{Z}$ are

$\displaystyle Y_{0}(x)=$ $\displaystyle 1$ $\displaystyle Y_{1}(x)=$
$\displaystyle x$ $\displaystyle Y_{2}(x)=$ $\displaystyle 2x^{2}-1$
$\displaystyle Y_{3}(x)=$ $\displaystyle 8x^{3}-5x$ $\displaystyle Y_{4}(x)=$
$\displaystyle 16x^{4}-16x^{2}+3$ $\displaystyle Y_{5}(x)=$ $\displaystyle
24x^{5}-28x^{3}+7x$ $\displaystyle Y_{6}(x)=$ $\displaystyle
16x^{6}-24x^{4}+10x^{2}-1.$

The polynomials $X_{5}(x)\pm 1$ and $Y_{5}(x)\pm 1$ are irreducible over
$\mathbb{Z}$ and none is a minimal polynomial of $\cos(\frac{2\pi}{d})$, of
which there are only two of degree $5$:

$\displaystyle\Psi_{11}(x)=$ $\displaystyle
32x^{5}+16x^{4}-32x^{3}-12x^{2}+6x+1$ $\displaystyle\Psi_{22}(x)=$
$\displaystyle 32x^{5}-16x^{4}-32x^{3}+12x^{2}+6x-1.$

These counter examples for $X_{n}(x)\pm 1$ and $Y_{n}(x)\pm 1$, show that, in
general, they do not have factorizations using the minimal polynomials of
$\cos(\frac{2\pi}{d})$.

##  6 Conclusion

We have solved the problem of factoring variants of Chebyshev polynomials of
the third and fourth kinds, $V_{n}(x)\pm 1$ and $W_{n}(x)\pm 1$, in terms of
minimal polynomials for $\cos(\frac{2\pi}{d})$. This was done by applying the
method of Wolfram [12] for factoring $T_{n}(x)\pm 1$ and $U_{n}(x)\pm 1$ in a
similar way. We have shown that there are no generalizations of this
factorization to similar variants of Chebyshev polynomials of the fifth and
sixth kinds, $X_{n}(x)\pm 1$ and $Y_{n}(x)\pm 1$.

## Acknowledgment

I am grateful to the College of Engineering & Computer Science at The
Australian National University for research support.

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