# Correlation of biological and computer viruses through evolutionary game
theory

Dimitris Kostadimas, Kalliopi Kastampolidou and Theodore Andronikos  Ionian
University
Department of Informatics
Corfu, Greece
Email: {p19kost2, c17kast<EMAIL_ADDRESS>
###### Abstract

Computer viruses have many similarities to biological viruses, and their
association may offer new perspectives and new opportunities in the effort to
tackle and even eradicate them. Evolutionary game theory has been established
as a useful tool for modeling viral behaviors. This work attempts to correlate
a well-known virus, namely Virlock, with the bacteriophage $\phi 6$.
Furthermore, the paper suggests certain efficient strategies and practical
ways that may reduce infection by Virlock and similar such viruses.

###### Index Terms:

Game theory, virus, computer virus, Virlock, biological systems, $\phi 6$,
evolutionary game theory.

##  I Introduction

###  I-A Computer viruses

The most common term that is currently being used to refer to malicious
computer programs is that of _computer virus_. The rationale behind this
metaphorical term being used to describe this kind of software is based in its
observed behavior that appears to have very much in common with the behavior
of biological viruses. Fred Cohen, the American computer scientist who coined
this term in 1983 described them as self-replicating programs that infect
other ones by embedding their ill-intended code in them [1].

Computer viruses, just like their biological counterparts, appear in many
different variations. Traits like their operational conditions, their host
demographic target, their replication manners, the infection type, the
infection success rate, the way they spread, the consequences and their
severity, and many other traits are what differentiate each and every computer
virus while many are named after these kinds of traits (like Ransomware for
example). Viruses are also usually classified based on the way they infect the
host machine [2], [3].

In the past, a computer virus was created for fun, leaving no permanent damage
to the host computer. However, this has changed. A common incentive behind
computer viruses is to bring profit to the owner of the malicious software, or
generally benefit him in some way. One of the ways this is achieved is by
stealing the infected host’s private data, encrypting them and asking the
victim (also known as the sucker) to pay a ransom in exchange for the key to
decrypt the encrypted data. It appears that recently there is an increase in
popularity of viruses that attack the privacy of computer users in order to
bring profit (illegally) to certain individuals. This calls for more ways and
layers of protection to be established.

As mentioned before, today’s computer viruses come with numerous traits and
are usually categorized based on their infection mechanics and behavior.
Viruses are part of the _malware_ family, just like _worms_ are [4]. The term
_malware_ refers to any kind of malicious software. Some common types of
malware are _ransomware, spyware, adware, trojan horses, rootkits, keyloggers_
, and of course more general types like viruses and worms [5], [6]. It is
worth noting that there are many more types of malware. For instance,
ransomware is a type of computer virus that encrypts the victim’s data and
asks them to pay a ransom in exchange for the decryption key or general access
back to the computer in case of a total lockdown [7].

Another trait that differentiates viruses is whether they contain/implement a
worm component in them or not. The worm component of a virus is what enables
them to spread through a network of computers, leading to reproduction of its
kind while also causing mutations to take place in the cases of polymorphic or
metamorphic code.

For a computer virus to engage for the first time with the host computer and
to start its replication, it must first infect the targeted computer. There
appears to be a certain amount of randomness in the way the virus continues to
spread in the cases of worms and viruses with a worm component and with a
polymorphic or metamorphic code. There are many ways for a malware to infect a
system. The most common ways are through email attachments, downloading
suspicious files, use of non-trustworthy removable media, security
vulnerabilities, P2P file sharing, malvertising (that is ads that promote
malicious software, found even in trustworthy sites), as well as through the
network due to the worm component of a virus (that got spread from another
infected computer) without the users taking any actions [4].

A usual virus would replicate itself by attaching its malicious code to
multiple host computer files as the time passes. Worms on the other hand do
not even need to manipulate the files in a computer in order to duplicate
themselves [8]. Worms remain active in the memory and the CPU of the computer
and their actions are usually invisible to the user except when they consume
enormous amounts of the computer’s resources, in which case the slow
performance will hint their existence. The most interesting thing regarding
worms and the viruses that contain a worm component, is not only that they are
able to replicate themselves in a rapid way, but mainly the fact that they can
do that without any human interaction with the computer whatsoever [9], [4],
[10].

In addition to replication manners and dynamics, some specific types of
viruses have the ability to mutate in order to achieve better success rates,
better spread, and generally enhance their already existing traits. This
ability is a trait of _polymorphic_ and _metamorphic_ viruses. More
specifically, a _polymorphic_ virus makes use of a variable encryption to
encrypt itself in order to make every copy of itself unique.

Another way virus mutants could appear is when the creator of the malicious
software has awareness of the virus’s state in the affected computers and its
effectiveness, and wants to make the malware more powerful and spread some new
variation instead. The creator might take the decision to alter the original
code of the virus and start spreading the new and updated mutant of the
original virus in order to make it more effective and accomplish ill-intended
goals.

The intent behind the use of this kind of polymorphic and metamorphic code is
to evade their detection from antimalware and antivirus software. Implementing
the “polymorphic trait” into a virus tends to be a fairly easier in comparison
to the “metamorphic trait.” However, the cost of the implementation might be
worth the extra effort as it also offers better protection against antivirus
software because it renders the virus way harder to be detected. In order to
prevent detection, some worms and viruses, especially the ones with a worm
component, manage to implement stealth strategies. Some hide themselves by not
taking up more space when replicating themselves by getting attached into the
host’s files. Other viruses attempt to kill processes run by active antivirus
software in the computer or the operating system to protect themselves and let
the user have a false sense of security.

###  I-B Biological viruses

Biological viruses are organisms that act parasitically and need to infect a
host in order to be able to reproduce and carry their genetic material, either
DNA or RNA and proteins. They cannot synthesize proteins and, therefore, use
host ribosomes to translate their RNA into proteins that serve them. Viruses
are transmitted in different ways, depending on their species. The number of
cells infected with a virus is called “host range.” The most prominent way of
dealing with a biological virus is the immune system of the organism whose
body it will infect. Usually the infected organisms are animals, plants,
molecules, and, of course, humans. Additionally, vaccines provide a good
defense and help the immune system, usually in regard to a specific virus
infection. Apart from vaccines, antiviral drugs are also available and are
evolving over time. However, there are some categories of viruses that attack
the organism’s immune system that they have infected, which cause chronic
infections.

When a cell is infected with a virus, it necessarily and directly replicates
itself in copies of the virus. Viruses are made up of their genetic material,
the capsid, a set of proteins that protect the genetic material, and in some
cases from external lipids. The virion is the extracellular form of the virus.
Depending on their genome, whether it is a DNA or an RNA genome, they are
called as such (DNA & RNA virus respectively). For an RNA virus, the genetic
material consists of ribonucleic acid (RNA) [11].

The effects that a virus has on an organism are numerous. Most cause the death
of the host cell. Usually, death involves restricting the normal activity of
the cell by viral proteins. The effects of some viruses can cause permanent
damage to the host organism or can be eliminated without malignancy. Some
viruses infect an organism and do not cause changes in the cells. Therefore,
their function continues normally with the virus, however ending up infecting
persistently eventually. Virome is the set of viruses that infects an
organism. Phage typing is a common method for tracing the source of infections
[12].

Precisely because viruses are acellular organisms, they are not transmitted by
cell division. For this purpose, they use the host, in order to create many
copies of themselves. When a virus infects a host, the host is forced to
reproduce the original virus. There is a basic life cycle for viruses.
Infection begins with the attachment of a virus and its proteins to the
surface of the host. At this point, the host range and cell type are
determined. This is followed by the penetration of virions into the cell.
Bacteria do not have a strong protective wall and viruses have developed
mechanisms for gene penetration, while the capsid remains outside the cell.
The final step is the release of the virus into the host cell by the Uncoating
process [13].

Replicating the virus also means multiplying the genome. After replication,
particles and altered proteins may appear relative to the original form of the
virus prior to penetration. Lysis is the process by which a virus is released
from a host cell. This causes the cell to be killed. Prophage is the process
by which the host reproduces, so the virus is also replicated. When the virus
ceases to be inactive, it causes a lysis in the host cell. Reproduction of an
RNA virus occurs in the cytoplasm. Each virus has its own enzymes that make
copies of the genomes. After lysis, the virus can infect another, new host
cell, leading to the repetition of this cycle. Also, during this step, there
may be mutations of the virus [14]. When an organism’s immune system is
exposed to a virus, it produces antibodies to suppress the virus. This process
is called humoral immunity. Depending on the antibodies that are produced, it
remains to be seen whether the body has recovered from the virus or not.

Bacteriophages or phages are viruses that alter or diverse microbial
populations. They were used as antibacterial agents due to their properties
[15]. The host range of some bacteriophages focuses on a single bacterial
strain. Bacteriophages are one of the groups of viruses and infect specific
bacteria. They usually have double-stranded RNA genomes.

An RNA virus consists of segments that make up a protein. These segments exist
in the capsid. Different segments may be in different virions and yet the
virus may be contagious. The way they do the infection is by attaching
themselves to molecules of the surface of the bacterion and then they enter
the cell. In many cases, as soon as the original virus enters the cell, it
begins to translate its mRNA into proteins. Then, the result of this process
either becomes virion and helps in the formation of other such virions or
participates in the process of cell lysis. Virus enzymes contribute to the
destruction of the cell membrane. The main way in which bacteria are protected
from such infections is with the help of enzymes that target unknown RNA.
Bacteria can also detect genomes from viruses that have fought in the past and
block their reproduction through RNA interference. This is a mechanism of
defense for bacteria against such infections. By their nature, bacteria have
the tactic of interfering RNA. During the replication of a viral RNA, some
mutations occur, which may either not affect the cell proteins or contribute
to a resistance to antiviral drugs.

###  I-C Evolutionary game theory

Evolutionary game theory (EGT) is a tool that helps with the modeling of the
behavior of this kind of viruses, thus setting a path on designing a higher
level of security. It has captured the interest of scientists such as
biologists, mathematicians, economists, psychologist, computer scientists and
many more. Its connections and applications to realistic real-life conditions
is what makes this concept even more captivating. EGT provides scientists with
the appropriate tools to study instinctive behaviors, biological phenomena and
even decisions based on rationality. The term population dynamics is now
widely used and refers to phenomena and behaviors just like the above, as well
as sets of strategies or characteristics players might inherit [16].

The microscopic level is just as interesting as the macroscopic kind of
observation. Even in biological processes, game properties have been observed.
Taking a deeper look into the cells and macromolecules that are part of
multicellular organisms, game properties can be observed. The strategy that
each one of them adopts is based on their moves. The strategy of a certain
player can (and probably will) change as they follow the principle of natural
selection. Strategies are always subject to change as throughout the cell’s
lifespan mutations tend to happen which cause irreversible changes, as well as
reversible changes caused by epigenetic modifications. All the above are
related to the reproduction of these objects and it is evident that
reproductive success alters the game’s outcome.

A brief overview of evolutionary games in the context of biological systems is
given in [17]. The incorporation of games in biology is a bigger persistent
trend. Many classical games, including the famous Prisoners’ Dilemma (see [18]
for references), have been used to model biological situations. This is not
limited to viruses, as it extends to microbes and their games (see [19]), and
even to bio-inspired models of computation (see [20] and [21] for details).
The introduction of unconventional tools, such as games, automata, notions
from quantum mechanics, promises to bring new perspectives and new insights to
the study of biological processes. For example, the adoption of concepts from
game theory to the field of quantum computation has proved to be extremely
successful (see [22], [23], [24] and [25] for some recent results and more
related references). It is worth pointing out that games may tackle critical
problems; coin tossing plays a crucial role in the design of quantum
cryptographic protocols (see [26] and references therein, and the more recent
[27]).

This paper offers a new perspective on the correlation of computer viruses to
biological viruses. There are of course many types of biological viruses, and
the same can be said about computer viruses. The behavioral traits of
biological viruses can be associated with the corresponding traits of the
computer viruses. The emphasis in this work is placed on a well-known computer
virus, namely Virlock, and its similarities with the biological virus $\phi
6$. These similarities, along with some of the anticipated differences, are
thoroughly examined and analyzed in Section V. We hope that this approach will
shed new light in the adoption and application of strategies that have been
successful in tackling viruses of one type to the other type, as well as
enhance the means to assess the effectiveness of the employed strategies.

##  II The Virlock virus

VirLock asks its victims to pay a ransom in order to regain access to their
files and their computing systems in general. VirLock has a parasitic
behavior. From the time it is executed, it starts infecting the supported
computer files, but the way it alters/infects a file is a bit different from
what normally happens with this type of malware. VirLock embeds clean code
inside a malware instead of malware inside clean code. This means that every
file that has been encrypted will be embedded into the malware. Now every
infected file can infect as it works like a mutation of VirLock. VirLock self-
replicates itself this way and grown its “population.”

The first detection of the VirLock virus happened in 2014 [28]. Of course, as
it is a polymorphic virus, many different mutations have been encountered
through the span of several years until today. As VirLock continues to evolve,
differences were found not only in the _decoration-code_ , but also in its
core functions. Specifically, the virus is also able to spread through
networks, thanks to the cloud storage that more and more people start using
nowadays. The mutations are many and there are multiple variables of VirLock
in the databases of several antivirus software and, as some results of the
famous website virustotal hint, the mutations of the virus probably helped
evade detection from certain antiviruses over time.

VirLock is able to occupy the whole screen area of the computer and kill the
_explorer.exe_ task of the Windows operating system that handles the graphical
user interface [29]. This means that it renders the infected computer almost
useless by the time it infects it, as there is no way to access the main
functions of the operating system because the whole screen is being occupied
by the virus message, while binary files and files with certain extensions are
being “encrypted” in the background. Because of this, the user has no way of
using an antivirus software the conventional way. As suggested by many
antivirus companies, the best way to try and disinfect a computer from VirLock
is to boot into the safe mode with network capabilities that Windows OS
offers. This way, VirLock probably will not be able to launch itself during
the startup. If the OS is booted successfully, then the user can perform a
virus scan with an antimalware or antivirus software to attempt virus
detection. The chances of an antivirus software detecting the virus of course
depend on what kind of VirLock variation this computer is infected with, since
it is quite possible that a new variant could be unrecognizable yet. Moreover,
there is a high chance that by the time the user acquires the knowledge of how
to proceed in the disinfection, the virus already has encrypted most, if not
all, of the computer files.

Every time VirLock encrypts a file, it appends the .exe extension to it and
renders it a copy of itself. So, every infected file is technically a
variation of the malware itself. Antivirus software are usually not able to
decrypt files, so the least they can do is detect the malware and quarantine
it or completely delete it. Deleting all the infected files in the computer is
obviously not an optimal solution. Some anti-virus software offer a VirLock
cleaner that is able to wipe the virus’ remnants and also “decrypt” most (if
not all) of the infected files in case of known VirLock variants. The user is
informed that false positives may also be found and should be careful during
the removal process. As VirLock has many variants, which makes it hard for
antivirus software to detect it, it is clear that What would help in the
detection of this kind of software is the study of its behavior.

Antivirus programs that are able to do live behavioral analysis have a
definite advantage in tackling the virus. This is because the mutations in
this case have something in common, and this is the core code. Even if the
code mutates as it self-replicates, the core functions remain the same. By
focusing on them and the way they react, it is possible to achieve a better
level of protection. Even though this is important, there are still ways that
VirLock dodges the emulations of antivirus software thanks to tricks like the
payload encryption and its obscure code in general. Even if security is much
more advanced nowadays, there’s still reportedly about 70% of malware that
manages to evade detection attempts performed by antivirus and antimalware
software [30], [31].

Security specialists have also found that VirLock appears to have an exploit
by itself. By entering 64 zeros in the _TrasnferID_ field, it is possible to
trick VirLock that the ransom has been paid. After that, by clicking a file,
the decrypting process is activated. The drawback of this strategy is that the
user will have to do this for every single file in the computer, with the risk
of infecting the computer once again. Another known exploit of this malware is
that it ignores the volume shadow copies of Windows. So, if this feature is
enabled and volume shadow copies of the computer are present, then the damage
can be reverted. Probably the best way to protect against this kind of viruses
is to hold constant backups of the precious files. Obviously, an updated
antivirus software and network segmentation may also prevent the spread of the
virus [32].

The latest variants of VirLock appear to be fairly powerful and effective,
even though enough time has passed since its first appearance and outbreak.
When VirLock infects a computer, it displays a message occupying the entire
screen area informing the user that pirated software was detected in the
computer. Thus, a fine must be paid for this illegal action in order to not
get arrested. To persuade the user that this notice is legit, VirLock has
localized GUI, so depending on the victim’s location it will display the
logotypes of the corresponding local authorities and government. Meanwhile,
the files of the infected computer are being encrypted in the background and
are also being infected. The term _encryption_ in this case deserves some
clarification [33].

VirLock does not use a one-way encryption algorithm like AES or RSA that some
of the more popular ransomware tend to use. Instead, it performs a two-stage
encryption, making use of the XOR and XOR-ROL operations [29]. Entropy is not
as high as if AES and RSA were used. This operation that makes the data appear
more obscure will still be referred to as encryption in the article. The
infection happens by trying to run the malicious file. When VirLock is
executed, it drops 3 randomly named executables in randomly named folders. As
the virus is polymorphic, these executables have identical hashes that are
different every time [34] and older variants appear to drop only 2 of them.
One of those registers itself as a Windows service to cover itself up, while
the others encrypt and infect the computer’s files. Other measurements that
VirLock takes to protect itself is that it disables the task manager process
so that the user is unable to take control of what is happening in the
computer and kill the virus that alters the Windows Registry. These are
considered to be VirLock’s trademark attributes. The first registry entry that
VirLock alters concerns the User Access Control (UAC). By disabling UAC, the
virus is able to manipulate everything in the computer freely without the need
of administration privileges. It then hides the known file extensions in order
to trick the user that the files are fine and making him unable to see the
.exe extension that VirLock appends, so the user might backup those files
thinking they are safe and moving them to another computer or try to run them
again. The last registry change is one that makes hidden files invisible [29].

Figure 1: Representation of VirLock Cloud Storage Infection. In the case
above, the bottom left computer is getting infected with the VirLock malware
through a malicious email attachment that the user opened. VirLock will infect
the files in the cloud storage as well, and the rest of the computers in the
network might eventually get the malware. The red arrows represent the path of
the infection, while the blue arrows represent interaction with the cloud.

The structure of this type of files is as follows, going from top to down: the
start and the end of the code is made out of polymorphic code that changes in
every iteration, so basically the code is wrapped around from this kind of
polymorphic code that we also refer to as decoration code. After the first
piece of polymorphic code, the malicious code, which runs every time, appears.
Right after the malicious code, VirLock embeds the clean code, which is
followed by the last piece of polymorphic code [33].

Of course, there is always the possibility that the user might be tempted to
pay the ransom instead of getting involved in any complicated task as they
engage in the virus removal process or taking advantage of the malware
exploits. There are sources claiming that a fair percentage of users proceed
on paying the ransom in order to regain access to their computer and data,
especially users under the age of 55.

Even though paying the ransom might be tempting for the infected users,
especially for companies with data of critical importance, there is an
important reason why one should think twice before proceeding with the
payment. First, there is a strong ethical reason, particularly in view of the
outrageous amount of money the ransomware may ask. Second, there are several
sources claiming that only 8% of those who proceed in the ransom payment
manage to get the entirety of their data back. It is also reported that “on
average, only 65% of the encrypted data was restored after the ransom was
paid.” Even if the victims are really desperate to get their data back, it
appears that there are other much more effective ways to achieve that.

##  III Modeling Virlock using game theory

Modeling the above situation with the help of game theory, a clearer picture
of the strategy that would benefit the user the most emerges. The following
payoff matrix is constructed keeping in mind the general consensus of security
specialists that “96% of those whose data was encrypted got their data back in
the most significant ransomware attack,” which confirms that there are other
more effective ways to retrieve the data besides paying the ransom.

Figure 2: Ransom Payment Payoff Matrix. The user has the option to pay and the
option not to pay the ransom, while the VirLock malware may or may not decrypt
the users’ data. The payoff matrix makes clear that the user will benefit the
malware creator/s less by not proceeding in the ransom payment, and that
paying the ransom holds an additional risk.

##  IV Pseudomonas virus $\phi 6$

Bacteriophage $\phi 6$ lytic virus belongs to the cystoviridae viruses and
aims to infect Pseudomonas bacteria. Its genome is double-stranded, consists
of three segmented parts and codes for 12 proteins. Such species consist of a
lipid membrane which surrounds their nucleocapsid.

$\phi 6$ locates and then sticks to the bacterium that wants to infect with
its special protein for this purpose, P3. Beyond that, different proteins
contribute to the process of cell infection. The bacteriophage $\phi 6$ has
been widely used to model its behavior and structure, and has previously been
associated with the field of classical and evolutionary game theory.

##  V Comparing the two viruses

A comparison between the way computer viruses and biological viruses operate
when they invade a host as well as their characteristics is of great interest
and could potentially provide new insights. Just like their biological
counterparts, computer virus types continue to evolve through the passage of
time following the evolution of computers. When a computer virus mutates,
generations can be observed during time-spans, just like when biological
viruses mutate.

Antivirus software could metaphorically be the immune system of a computer.
Computer viruses try to weaken this system in order to replicate themselves
and grow their population by invading the victim’s computer files (which in
this case could represent the cells of a human organism) and, as an extension,
the whole network of computers connected with the original victim.

The main properties and characteristics of VirLock that could be linked to
bacteriophages and especially $\phi 6$ are the following:

  1. 1.

They are self-replicating viruses that appear to grow exponentially.

  2. 2.

They try to protect themselves by attacking and eventually manipulating the
host.

  3. 3.

They affect the host in order to gain full access to its functions and keep
their viral ability unaffected.

  4. 4.

They affect certain host types.

  5. 5.

They exhibit parasitic behavior, while manipulating and embedding their code
in their replicants/mutants.

  6. 6.

They are able to spread when the infected parts come in contact with other
hosts.

  7. 7.

They have a core structure that are subject to change.

  8. 8.

They are able to mutate rapidly not only by themselves but also with external
help.

  9. 9.

They can be untraceable for a specific time, they are, in general, hard to
locate and extremely difficult to eliminate.

Figure 3: Similarities between the two viruses. Figure 4: Differences between
the two viruses.

The following tables describe the steps of some of the known strategies that
users could follow in order to recover their computer back to a normal
functioning state. The rationale behind the following tables is to get an
overall sense of the complexity inherent in every strategy, as well as the
effectiveness and the risks that one has to take.

Figure 5: Complexity of the steps taken in every recovery strategy. Figure 6:
Ransom payment. Figure 7: Decryption taking advantage of VirLock’s exploit.
Figure 8: Recovery using shadow volume copies. Figure 9: Simple malware
removal (with antivirus software). Figure 10: Virus removal &
cleaner/recoverer (antivirus & cleaner).

The _complexity_ variable has a range from 0 to 10 and is based on how complex
it would be for an average computer user to perform one of those actions. The
_effectiveness_ variable can take 3 different values (either low, medium or
high) and it is based on the success rate that this technique has, as well as
the percentage of the recovered files (in case where the entirety of them is
not recovered). The risk of _re-infection_ variable takes the same values with
the effectiveness variable and indicates whether the user is highly
susceptible to get infected again or not, while using a certain recovery
strategy. The strategies are also broken down into steps and a complexity
value is also picked for each step with an average computer user in mind.

These tables (as well as the ransom payment payoff matrix) will help users
infected with the VirLock malware adopt an optimal strategy for their
infection scenario. The tables might also be useful to users with infections
from similar malware. Having a well-thought-out plan (like those tables offer)
beforehand can offer the user a clear advantage.

The effectiveness of the recovery using shadow volumes strategy is high, but
this depends on whether the user has kept any of these copies beforehand and
how old these copies are. The “click on every file of your computer” step
depends on how many files the users stores in the computer. Installing an
antivirus or anti-malware software is not always required, as one might have
already been installed beforehand.

##  VI Conclusion and further work

Computer viruses can be classified into several categories depending on their
characteristics. However, their similarity in relation to biological viruses
is quite evident in many properties, as long as one correlates the appropriate
viruses with each other, depending on their common behavioral elements.
Studying them and the correlation between computer viruses and biological
viruses offers an alternative look and approach on how to deal with both
biological and computer viruses. It would also be interesting to identify
unique strategies of one group and try to apply or simulate them in the other
group, enforcing the appropriate parallelism, in order for new insights and
solutions to arise.

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# Some identities on generalized harmonic numbers and generalized harmonic
functions

Dae San Kim  Department of Mathematics, Sogang University, Seoul 121-742,
Republic of Korea<EMAIL_ADDRESS>,  Hyekyung Kim  Department Of
Mathematics Education, Daegu Catholic University, Gyeongsan 38430, Republic of
Korea<EMAIL_ADDRESS>and  Taekyun Kim  Department of Mathematics, Kwangwoon
University, Seoul 139-701, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

The harmonic numbers and generalized harmonic numbers appear frequently in
many diverse areas such as combinatorial problems, many expressions involving
special functions in analytic number theory and analysis of algorithms. The
aim of this paper is to derive some identities involving generalized harmonic
numbers and generalized harmonic functions from the beta functions
$F_{n}(x)=B(x+1,n+1),\,\,(n=0,1,2,\dots)$ using elementary methods.

###### Key words and phrases:

generalized harmonic numbers of order $\alpha$; generalized harmonic function

###### 2010 Mathematics Subject Classification:

11B68; 11B83; 26A42

##  1\. Introduction

For $s\in\mathbb{C}$, the Riemann zeta function is defined by

$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}},\quad(\mathrm{Re}(s)>1),\quad(\mathrm{see}\
[1,12,14]).$

As is well known, we have

(1)
$\zeta(2n)=\frac{(-1)^{n+1}B_{2n}}{2(2n)!}(2\pi)^{2n},\quad(n\in\mathbb{N}),\quad(\mathrm{see}\
[14]),$

where $B_{n}$ are the Bernoulli numbers defined by

(2)
$\frac{t}{e^{t}-1}=\sum_{n=0}^{\infty}B_{n}\frac{t^{n}}{n!}=1-\frac{1}{2}x+\frac{1}{6}\frac{x^{2}}{2!}-\frac{1}{30}\frac{x^{4}}{4!}+\frac{1}{42}\frac{x^{6}}{6!}-\frac{1}{30}\frac{x^{8}}{8!}+\cdots,\quad(\mathrm{see}\
[1-14]).$

Thus we see from (1) and (2) that

(3)
$\zeta(2)=\frac{\pi^{2}}{6},\quad\zeta(4)=\frac{\pi^{4}}{90},\quad\zeta(6)=\frac{\pi^{6}}{945},\quad\zeta(8)=\frac{\pi^{8}}{9450},\cdots.$

More generally, Hurwitz zeta function is defined by

(4)
$\zeta(x,s)=\sum_{n=0}^{\infty}\frac{1}{(n+x)^{s}},\quad(\mathrm{Re}(s)>1,x>0),\quad(\mathrm{see}\
[14]).$

The harmonic numbers are defined by

(5) $H_{0}=0,\quad
H_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n},\quad(n\in\mathbb{N}),\quad(\mathrm{see}\
[6-8,10,13]).$

The generating function of harmonic numbers is given by

(6) $-\frac{1}{1-x}\log(1-x)=\sum_{n=1}^{\infty}H_{n}x^{n},\quad(\mathrm{see}\
[6-8,13]).$

For $\alpha\in\mathbb{N}$, the generalized harmonic numbers of order $\alpha$
are defined by

(7) $H_{0}^{(\alpha)}=0,\quad
H_{n}^{(\alpha)}=1+\frac{1}{2^{\alpha}}+\frac{1}{3^{\alpha}}+\cdots+\frac{1}{n^{\alpha}},\quad(\mathrm{see}\
[3-5,8]).$

For $s\in\mathbb{C}$ with $\mathrm{Re}(s)>0$, the gamma function is given by

(8) $\Gamma(s)=\int_{0}^{\infty}e^{-t}t^{s-1}dt,\quad(\mathrm{see}\
[9,12,14]),$

with $\Gamma(s+1)=s\Gamma(s)$ and $\Gamma(1)=1$. For $\mathrm{Re}(\alpha)>0$
and $\mathrm{Re}(\beta)>0$, the beta function is given by

(9)
$B(\alpha,\beta)=\int_{0}^{1}t^{\alpha-1}(1-t)^{\beta-1}dt,\quad(\mathrm{see}\
[9,12,14]).$

From (8) and (9), we have

(10)
$B(\alpha,\beta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)},\quad(\mathrm{see}\
[14]).$

Lastly, the following binomial inversion theorem is well known.

######  Theorem 1.1.

For any integer $n\geq 0$, we have

$b_{n}=\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}a_{k}\Longleftrightarrow
a_{n}=\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}b_{k}.$

In this paper, we derive some expressions for
$\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{1}{(k+x+1)^{r}},\quad\zeta(x+1,r)$,
and $\,\,r!$ in terms of generalized harmonic functions and generalized
harmonic numbers using elementary methods like the binomial inversion,
differentiation and integration.

##  2\. Some identities on generalized harmonic numbers and generalized
harmonic functions

For every $\alpha\in\mathbb{N}$, we consider the generalized harmonic function
$H_{n}(x,\alpha)$ defined by

(11) $H_{n}(x,\alpha)=\sum_{k=0}^{n}\frac{1}{(k+x+1)^{\alpha}},\quad(x\geq
0),\quad(\mathrm{see}\ [3-5]).$

Note that

(12) $\frac{d}{dx}H_{n}(x,\alpha)=-\alpha H_{n}(x,\alpha+1).$

In particular, for $x=0$, we have $H_{n}(0,\alpha)=H_{n+1}^{(\alpha)}$.

For $x>-1$ and $n=0,1,2,\dots$, we let

(13) $\displaystyle F_{n}(x)$
$\displaystyle=\int_{0}^{1}(1-t)^{n}t^{x}dt=B(x+1,n+1)=\frac{\Gamma(x+1)\Gamma(n+1)}{\Gamma(x+n+2)}$
$\displaystyle=\frac{n!}{(x+n+1)(x+n)\cdots(x+1)}=\frac{1}{\binom{x+n+1}{n}(x+1)}.$

By binomial theorem, (13) is also equal to

(14) $\displaystyle F_{n}(x)=\int_{0}^{1}(1-t)^{n}t^{x}dt$
$\displaystyle=\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\int_{0}^{1}t^{k+x}dt$
$\displaystyle=\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{1}{x+k+1}.$

We will use the following lemma repeatedly throughout this paper.

######  Lemma 2.1.

Let $F_{n}(x)=B(x+1,n+1)$. Then, for any positive integer $r$, we have

$\displaystyle(a)\quad F_{n}^{(r)}(x)=\int_{0}^{1}(1-t)^{n}(\log
t)^{r}t^{x}dt=r!\sum_{k=0}^{n}\binom{n}{k}(-1)^{k-r}\frac{1}{(x+k+1)^{r+1}},$
$\displaystyle(b)\quad F_{n}^{\prime}(x)=-H_{n}(x,1)F_{n}(x),$
$\displaystyle(c)\quad\sum_{n=1}^{\infty}\frac{1}{n}\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{1}{(k+1)^{r}}=r.$

###### Proof.

(a) follows from (14).
(b) From (13), we get

$\displaystyle F_{n}^{\prime}(x)$
$\displaystyle=\frac{d}{dx}F_{n}(x)=\frac{d}{dx}\bigg{(}\frac{n!}{(x+n+1)(x+n)\cdots(x+1)}\bigg{)}$
$\displaystyle=-\sum_{k=0}^{n}\frac{1}{k+x+1}\cdot\frac{n!}{(x+n+1)(x+n)\cdots(x+1)}$
$\displaystyle=-H_{n}(x,1)F_{n}(x).$

(c) The left hand side of (c) is equal to

$\displaystyle\sum_{n=1}^{1}$
$\displaystyle\frac{1}{n}\int_{0}^{1}\cdots\int_{0}^{1}(1-x_{1}x_{2}\cdots
x_{r})^{n}dx_{1}dx_{2}\cdots dx_{r}$
$\displaystyle=-\int_{0}^{1}\cdots\int_{0}^{1}(\log x_{1}+\cdots+\log
x_{r})\,dx_{1}dx_{2}\cdots dx_{r}=r,$

where the integrals are $r$-multiple ones. ∎

From Lemma 2.1, (a), (b), and (13), we have

(15)
$\displaystyle\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{1}{(k+x+1)^{2}}=H_{n}(x,1)F_{n}(x)=\frac{H_{n}(x,1)}{\binom{x+n+1}{n}(x+1)}.$

In particular, for $x=0$, we get

(16)
$\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{1}{(k+1)^{2}}=\frac{1}{n+1}H_{n+1},\quad(n\geq
0),$

which, by Lemma 2.1 (c), yields

$\sum_{n=1}^{\infty}\frac{1}{n(n+1)}H_{n+1}=2.$

Therefore, from (15) and (16), by binomial inversion we obtain the following
theorem.

######  Theorem 2.2.

For $n\geq 0$, we have

$\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{H_{k}(x,1)}{\binom{x+k+1}{k}(x+1)}=\frac{1}{(n+x+1)^{2}},\quad\mathrm{and}\quad\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{H_{k+1}}{k+1}=\frac{1}{(n+1)^{2}}.$

We note from Theorem 2.2, (3) and (4) that

$\displaystyle\sum_{n=0}^{\infty}\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{H_{k}(x,1)}{\binom{x+k+1}{k}(x+1)}=\sum_{n=0}^{\infty}\frac{1}{(n+x+1)^{2}}=\zeta(x+1,2),$
$\displaystyle\sum_{n=0}^{\infty}\bigg{(}\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{H_{k+1}}{k+1}\bigg{)}=\sum_{n=0}^{\infty}\frac{1}{(n+1)^{2}}=\frac{\pi^{2}}{6}.$

From Lemma 2.1 (b), (12) and (13), we note that

(17) $\displaystyle F_{n}^{(2)}(x)$
$\displaystyle=\frac{d^{2}}{dx^{2}}F_{n}(x)=(-1)\frac{d}{dx}\Big{(}H_{n}(x,1)F_{n}(x)\Big{)}$
$\displaystyle=(-1)^{2}H_{n}(x,2)F_{n}(x)+(-1)^{2}\Big{(}H_{n}(x,1)\Big{)}^{2}F_{n}(x),$
$\displaystyle=(-1)^{2}\Big{(}H_{n}(x,2)+\big{(}H_{n}(x,1)\big{)}^{2}\Big{)}\frac{1}{\binom{x+n+1}{n}(x+1)},$

and that

(18) $\displaystyle F_{n}^{(3)}(x)$
$\displaystyle=\frac{d^{3}}{dx^{3}}F_{n}(x)=\frac{d}{dx}\Big{(}(-1)^{2}H_{n}(x,2)F_{n}(x)+(-1)^{2}\big{(}H_{n}(x,1)\big{)}^{2}F_{n}(x)\Big{)}$
$\displaystyle=2(-1)^{3}H_{n}(x,3)F_{n}(x)+(-1)^{3}H_{n}(x,2)H_{n}(x,1)F_{n}(x)$
$\displaystyle+2(-1)^{3}H_{n}(x,1)H_{n}(x,2)F_{n}(x)+(-1)^{3}\Big{(}H_{n}(x,1)\Big{)}F_{n}(x)$
$\displaystyle=2(-1)^{3}H_{n}(x,3)F_{n}(x)+3(-1)^{3}H_{n}(x,1)H_{n}(x,2)F_{n}(x)+(-1)^{3}\Big{(}H_{n}(x,1)\Big{)}^{3}F_{n}(x)$
$\displaystyle=(-1)^{3}\Big{(}2H_{n}(x,3)+3H_{n}(x,1)H_{n}(x,2)+\big{(}H_{n}(x,1)\big{)}^{3}\Big{)}\frac{1}{\binom{x+n+1}{n}(x+1)}.$

By Lemma 2.1 (a), we get

(19)
$F_{n}^{(2)}(x)=2!\sum_{k=0}^{n}\binom{n}{k}(-1)^{k-2}\frac{1}{(x+k+1)^{3}},\quad
F_{n}^{(3)}(x)=3!\sum_{k=0}^{n}\binom{n}{k}(-1)^{k-3}\frac{1}{(x+k+1)^{4}}.$

Therefore, by (17)-(19), we obtain the following theorem.

######  Theorem 2.3.

For $n\geq 0$, we have

$\frac{H_{n}(x,2)+(H_{n}(x,1))^{2}}{\binom{x+n+1}{n}(x+1)}=2!\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{1}{(x+k+1)^{3}},$

and

$\frac{2H_{n}(x,3)+3H_{n}(x,1)H_{n}(x,2)+(H_{n}(x,1))^{3}}{\binom{x+n+1}{n}(x+1)}=3!\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{1}{(x+k+1)^{4}}.$

For $x=0$, we also have

$2!\sum_{k=0}^{n}\binom{n}{k}\frac{(-1)^{k}}{(k+1)^{3}}=\frac{H_{n+1}^{(2)}+(H_{n+1})^{2}}{n+1},$

and

$3!\sum_{k=0}^{n}\frac{\binom{n}{k}(-1)^{k}}{(k+1)^{4}}=\frac{2H_{n+1}^{(3)}+3H_{n+1}H_{n+1}^{(2)}+(H_{n+1})^{3}}{n+1}.$

From Theorem 2.3, by binomial inversion we obtain

(20)
$\displaystyle\frac{1}{2!}\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{H_{k}(x,2)+\big{(}H_{k}(x,1)\big{)}^{2}}{\binom{x+k+1}{k}(x+1)}=\frac{1}{(x+n+1)^{3}},$
(21)
$\displaystyle\frac{1}{3!}\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\bigg{(}\frac{2H_{k}(x,3)+3H_{k}(x,1)H_{k}(x,2)+(H_{k}(x,1))^{3}}{\binom{x+k+1}{k}(x+1)}\bigg{)}=\frac{1}{(x+n+1)^{4}}.$

By (20) and (4), we get

$\frac{1}{2!}\sum_{n=0}^{\infty}\sum_{k=0}^{n}\frac{\binom{n}{k}(-1)^{k}}{\binom{x+k+1}{k}(x+1)}\Big{(}H_{k}(x,2)+\big{(}H_{k}(x,1)\big{)}^{2}\Big{)}=\zeta(x+1,3).$

For $x=0$, we have

$\frac{1}{2!}\sum_{n=0}^{\infty}\sum_{k=0}^{n}\frac{\binom{n}{k}(-1)^{k}}{k+1}\Big{(}H_{k+1}^{(2)}+(H_{k+1})^{2}\Big{)}=\zeta(3).$

In addition, from (21) and (4), we also have

$\frac{1}{3!}\sum_{n=0}^{\infty}\sum_{k=0}^{n}\frac{\binom{n}{k}(-1)^{k}}{\binom{x+k+1}{k}(x+1)}\Big{(}2H_{k}(x,3)+3H_{k}(x,1)H_{k}(x,2)+\big{(}H_{k}(x,1)\big{)}^{3}\Big{)}=\zeta(x+1,4).$

For $x=0$, from (3) we get

$\frac{1}{3!}\sum_{n=0}^{\infty}\sum_{k=0}^{n}\frac{\binom{n}{k}(-1)^{k}}{k+1}\Big{(}2H_{k+1}^{(3)}+3H_{k+1}H_{k+1}^{(2)}+(H_{k+1})^{3}\Big{)}=\sum_{n=0}^{\infty}\frac{1}{(n+1)^{4}}=\frac{\pi^{4}}{90}.$

From Theorem 2.3 and Lemma 2.1 (c), we note that

(22)
$\frac{1}{2!}\sum_{n=1}^{\infty}\frac{H_{n+1}^{(2)}+(H_{n+1})^{2}}{n(n+1)}=\sum_{n=1}^{\infty}\frac{1}{n}\sum_{k=0}^{n}\binom{n}{k}\frac{(-1)^{k}}{(k+1)^{3}}=3,$

and that

(23)
$\frac{1}{3!}\sum_{n=1}^{\infty}\frac{2H_{n+1}^{(3)}+3H_{n+1}H_{n+1}^{(2)}+(H_{n+1})^{3}}{n(n+1)}=\sum_{n=1}^{\infty}\frac{1}{n}\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{1}{(k+1)^{4}}=4.\\\
$

Therefore, by (22) and (23), we obtain the following corollary.

######  Corollary 2.4.

For $n\geq 1$, we have

$\sum_{n=1}^{\infty}\frac{H_{n+1}^{(2)}+(H_{n+1})^{2}}{n(n+1)}=3!,\quad\sum_{n=1}^{\infty}\frac{2H_{n+1}^{(3)}+3H_{n+1}H_{n+1}^{(2)}+(H_{n+1})^{3}}{n(n+1)}=4!.$

From (21), Lemma 2.1 (b), (12) and (13), we have

(24) $\displaystyle F_{n}^{(4)}(x)$
$\displaystyle=\frac{d}{dx}\Big{(}2(-1)^{3}H_{n}(x,3)F_{n}(x)+3(-1)^{3}H_{n}(x,1)H_{n}(x,2)F_{n}(x)+(-1)^{3}(H_{n}(x,1))^{3}F_{n}(x)\Big{)}$
$\displaystyle=6(-1)^{4}H_{n}(x,4)F_{n}(x)+2(-1)^{4}H_{n}(x,3)H_{n}(x,1)F_{n}(x)+3(-1)^{4}H_{n}(x,2)H_{n}(x,2)F_{n}(x)$
$\displaystyle+6(-1)^{4}H_{n}(x,1)H_{n}(x,3)F_{n}(x)+3(-1)^{4}\big{(}H_{n}(x,1)\big{)}^{2}H_{n}(x,2)F_{n}(x)$
$\displaystyle+3(-1)^{4}\big{(}H_{n}(x,1)\big{)}^{2}H_{n}(x,2)F_{n}(x)+(-1)^{4}\big{(}H_{n}(x,1)\big{)}^{4}F_{n}(x)$
$\displaystyle=(-1)^{4}\Big{(}6H_{n}(x,4)+8H_{n}(x,3)H_{n}(x,1)+3\big{(}H_{n}(x,2)\big{)}^{2}+6\big{(}H_{n}(x,1)\big{)}^{2}H_{n}(x,2)$
$\displaystyle+\big{(}H_{n}(x,1)\big{)}^{4}\Big{)}F_{n}(x)$
$\displaystyle=(-1)^{4}\frac{6H_{n}(x,4)+8H_{n}(x,3)H_{n}(x,1)+3(H_{n}(x,2))^{2}+6(H_{n}(x,1))^{2}H_{n}(x,2)+(H_{n}(x,1))^{4}}{\binom{x+n+1}{n}(x+1)},$

and

(25)
$F_{n}^{(4)}(0)=(-1)^{4}\frac{6H_{n+1}^{(4)}+8H_{n+1}^{(3)}H_{n+1}+3(H_{n+1}^{(2)})^{2}+6(H_{n+1})^{2}H_{n+1}^{(2)}+(H_{n+1})^{4}}{n+1}.$

On the other hand, by Lemma 2.1 (a), we get

(26)
$F_{n}^{(4)}(x)=4!\sum_{k=0}^{n}\binom{n}{k}(-1)^{k-4}\frac{1}{(k+x+1)^{5}}=\int_{0}^{1}(1-t)^{n}(\log
t)^{4}t^{x}dt,$

and

(27)
$F_{n}^{(4)}(0)=4!\sum_{k=0}^{n}\binom{n}{k}(-1)^{k-4}\frac{1}{(k+1)^{5}}=\int_{0}^{1}(1-t)^{n}(\log
t)^{4}dt.$

By (24)-(27), we get

(28)
$\displaystyle\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{1}{(k+x+1)^{5}}=\frac{1}{4!}\int_{0}^{1}(1-t)^{n}(\log
t)^{4}t^{x}dt$
$\displaystyle=\frac{6H_{n}(x,4)+8H_{n}(x,3)H_{n}(x,1)+3(H_{n}(x,2))^{2}+6(H_{n}(x,1))^{2}H_{n}(x,2)+(H_{n}(x,1))^{4}}{4!\binom{x+n+1}{n}(x+1)},$

and

(29)
$\displaystyle\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{1}{(k+1)^{5}}=\frac{1}{4!}\int_{0}^{1}(1-t)^{n}(\log
t)^{4}dt$
$\displaystyle=\frac{6H_{n+1}^{(4)}+8H_{n+1}^{(3)}H_{n+1}+3(H_{n+1}^{(2)})^{2}+6(H_{n+1})^{2}H_{n+1}^{(2)}+(H_{n+1})^{4}}{4!(n+1)}.$

Therefore, from (28) and (29), by binomial inversion we obtain the following
theorem.

######  Theorem 2.5.

For $n\geq 0$, we have

$\displaystyle\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{6H_{k}(x,4)+8H_{k}(x,3)H_{k}(x,1)+3(H_{k}(x,2))^{2}+6(H_{k}(x,1))^{2}H_{k}(x,2)+(H_{k}(x,1))^{4}}{4!\binom{x+k+1}{k}(x+1)}$
$\displaystyle=\frac{1}{(n+x+1)^{5}},$

and

$\frac{1}{4!}\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{6H_{k+1}^{(4)}+8H_{k+1}^{(3)}H_{k+1}+3(H_{k+1}^{(2)})^{2}+6(H_{k+1})^{2}H_{k+1}^{(2)}+(H_{k+1})^{4}}{k+1}=\frac{1}{(n+1)^{5}}.$

By Theorem 2.5 and (4), we have

$\displaystyle\sum_{n=0}^{\infty}\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{6H_{k}(x,4)+8H_{k}(x,3)H_{k}(x,1)+3(H_{k}(x,2))^{2}+6(H_{k}(x,1))^{2}H_{k}(x,2)+(H_{k}(x,1))^{4}}{4!\binom{x+k+1}{k}(x+1)}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{1}{(n+x+1)^{5}}=\zeta(x+1,5).$

For $x=0$, we also have

$\displaystyle\frac{1}{4!}\sum_{n=0}^{\infty}\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{6H_{k+1}^{(4)}+8H_{k+1}^{(3)}H_{k+1}+3(H_{k+1}^{(2)})^{2}+6(H_{k+1})^{2}H_{k+1}^{(2)}+(H_{k+1})^{4}}{k+1}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{1}{(n+1)^{5}}=\zeta(5).$

From (29) and Lemma 2.1 (c), we note that

$\displaystyle\frac{1}{4!}\sum_{n=1}^{\infty}\frac{6H_{n+1}^{(4)}+8H_{n+1}^{(3)}H_{n+1}+3(H_{n+1}^{(2)})^{2}+6(H_{n+1})^{2}H_{n+1}^{(2)}+(H_{n+1})^{4}}{n(n+1)}$
$\displaystyle=\sum_{n=1}^{\infty}\frac{1}{n}\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{1}{(k+1)^{5}}=5.$

Hence, we have

$\displaystyle\sum_{n=1}^{\infty}\frac{6H_{n+1}^{(4)}+8H_{n+1}^{(3)}H_{n+1}+3(H_{n+1}^{(2)})^{2}+6(H_{n+1})^{2}H_{n+1}^{(2)}+(H_{n+1})^{4}}{n(n+1)}=5!.$

From Lemma 2.1 (a), (b), and (12), we note that

$\displaystyle(r+1)!\sum_{k=0}^{n}\binom{n}{k}(-1)^{k-r}\frac{1}{(k+x+1)^{r+2}}=\frac{d^{r}}{dx^{r}}(H_{n}(x,1)F_{n}(x))$
$\displaystyle=\sum_{l=0}^{r}\binom{r}{l}\bigg{(}\frac{d^{l}}{dx^{l}}H_{n}(x,1)\bigg{)}\bigg{(}\frac{d^{r-l}}{dx^{r-l}}F_{n}(x)\bigg{)}$
$\displaystyle=\sum_{l=0}^{r}\binom{r}{l}l!(-1)^{l}H_{n}(x,l+1)\int_{0}^{1}(1-t)^{n}\big{(}\log
t\big{)}^{r-l}t^{x}dt,$

which gives us

(30) $\displaystyle\sum_{k=0}^{n}$
$\displaystyle\binom{n}{k}(-1)^{k}\frac{1}{(k+x+1)^{r+2}}.$
$\displaystyle=\frac{(-1)^{r}}{(r+1)!}\sum_{l=0}^{r}\binom{r}{l}l!(-1)^{l}H_{n}(x,l+1)\int_{0}^{1}(1-t)^{n}\big{(}\log
t\big{)}^{r-l}t^{x}dt.$

From (30), by binomial inversion and summing over $n$, we obtain

(31) $\displaystyle\frac{(-1)^{r}}{(r+1)!}$
$\displaystyle\sum_{n=0}^{\infty}\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\sum_{l=0}^{r}\binom{r}{l}l!(-1)^{l}H_{k}(x,l+1)\int_{0}^{1}(1-t)^{k}\big{(}\log
t\big{)}^{r-l}t^{x}dt$
$\displaystyle=\sum_{n=0}^{\infty}\frac{1}{(n+x+1)^{r+2}}=\zeta(x+1,r+2),$

which, for $x=0$, yields

$\displaystyle\frac{(-1)^{r}}{(r+1)!}$
$\displaystyle\sum_{n=0}^{\infty}\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\sum_{l=0}^{r}\binom{r}{l}l!(-1)^{l}H_{k+1}^{(l+1)}\int_{0}^{1}(1-t)^{k}\big{(}\log
t\big{)}^{r-l}dt$
$\displaystyle=\sum_{n=0}^{\infty}\frac{1}{(n+1)^{r+2}}=\zeta(r+2).$

By (30) with $x=0$ and Lemma 2.1 (c), we have

(32) $\displaystyle\frac{(-1)^{r}}{(r+1)!}$
$\displaystyle\sum_{l=0}^{r}\binom{r}{l}l!(-1)^{l}\sum_{n=1}^{\infty}\frac{1}{n}H_{n+1}^{(l+1)}\int_{0}^{1}(1-t)^{n}(\log
t)^{r-l}dt$
$\displaystyle=\sum_{n=1}^{\infty}\frac{1}{n}\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{1}{(k+1)^{r+2}}=r+2.$

Thus, from (30)-(32), we obtain the following theorem.

######  Theorem 2.6.

For $n,r\geq 0$, we have the following identities:

$\displaystyle\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{1}{(k+x+1)^{r+2}}=\frac{(-1)^{r}}{(r+1)!}\sum_{l=0}^{r}\binom{r}{l}l!(-1)^{l}H_{n}(x,l+1)\int_{0}^{1}(1-t)^{n}\big{(}\log
t\big{)}^{r-l}t^{x}dt,$
$\displaystyle\frac{(-1)^{r}}{(r+1)!}\sum_{n=0}^{\infty}\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\sum_{l=0}^{r}\binom{r}{l}l!(-1)^{l}H_{k}(x,l+1)\int_{0}^{1}(1-t)^{k}\big{(}\log
t\big{)}^{r-l}t^{x}dt$ $\displaystyle=\zeta(x+1,r+2),$

and

$\sum_{l=0}^{r}\binom{r}{l}l!(-1)^{l}\sum_{n=1}^{\infty}\frac{1}{n}H_{n+1}^{(l+1)}\int_{0}^{1}(1-t)^{n}(\log
t)^{r-l}dt=(-1)^{r}(r+2)!.$

##  3\. Conclusion

In this paper, we used elementary methods in order to derive some identities
involving generalized harmonic numbers and generalized harmonic functions from
the beta functions $F_{n}(x)=B(x+1,n+1),\,\,(n=0,1,2,\dots)$. In more detail,
for every $r=2,3,4,5$, we showed that

(33)
$\sum_{k=0}^{n}\binom{n}{k}(-1)^{k}\frac{1}{(k+x+1)^{r}},\quad\zeta(x+1,r),\quad
r!$

are all expressed in terms of generalized harmonic functions and generalized
harmonic numbers. The methods employed in this paper could be continued
further for $r=6,7,8,\dots$, which requires to find an explicit expression for
$F_{n}^{(r-1)}(x)$ in terms of the generalized harmonic functions. In Theorem
2.6, for any $r=2,3,4,\dots$, we found expressions for (33) without deriving
the explicit expressions for $F_{n}^{(r-l)}(x)=\int_{0}^{1}(1-t)^{n}(\log
t)^{r-l}dt$ in terms of the generalized harmonic functions.

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