Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

k-Order Mersenne Numbers

Mustafa Asci, Suleyman Aydinyuz*

Pamukkale University, Department of Mathematics, Denizli, Turkey

Abstract: In this paper, we define k-order Mersenne numbers inspired the Mersenne numbers. Then, we give some interesting properties such as generating function and sum of the first m terms of these numbers. Also, we give excellent results of the k-order Mersenne numbers. We introduce the matrix representation and obtain the k-order Mersenne matrix.

Keywords: Mersenne numbers, k-order Mersenne numbers, Generating function, Matrix representation

Received October 17, 2021; Revised November 20, 2021; Accepted December 01, 2021

Citation: Mustafa Asci, Suleyman Aydinyuz, k-Order Mersenne Numbers. Turkish Journal of Analysis and Number Theory, 9 (3). 48-51. January 2021. doi:10.12691/tjant-9-3-3

1. Introduction

The field of study, which constitutes an important part of number theory, is number sequences given by recurrence relations. Recurrence number sequences are of great importance in the advancement of many fields that constitute the main topics of science. Such sequences are frequently used in approach theory, which is an important area of numerical analysis, in cipher science, computer graphics and time series analysis in [1,2]. Today, there are many sources that gather and present the information obtained for linear recurrence relation.

Many researchers have studied well-known number sequences as Fibonacci, Lucas, Pell, Jacobsthal and Mersenne numbers to present their theories and applications in many fields such physics, engineering, computer science, natüre and arts. The generalizatons and applications of these numbers sequences given above are given in [3-12].

Mersenne numbers are numbers that are in mathematics minus the Powers of two anda re calculated as for the natural number . It is named after the French Mathematician, philosopher, monk and music theorist Marin Mersenne, known as the “Father of Accoustics”. We can say that Mersenne numbers belong to the same family as Fermat numbers and therefore have the same properties. Many studies have been done on these numbers and they are widely used in many fields such as encryption, computers, geometry and physics in [13,14,15,16,17].

The Mersenne numbers are defined as follows

or

with the initial conditions and in [9].

The first few terms of the sequence can be seen in the following Table 1

Table 1. The first few terms of the sequence Mersenne numbers

Proposition 1.1. If is the term of the Mersenne sequence in [9] then

1.

2. .

Proposition 1.2. The terms of Mersenne sequence with a negative index are given in [16] by

In this paper, we define the k-order Mersenne numbers and give some properties about these numbers. We also define the matrix representation for Mersenne numbers of order k and define the Q-matrix introduced fort he Fibonacci matrix for k-order Mersenne numbers.

2. k-Order Mersenne Numbers

Definition 2.1. Let k be an integer. The k-order Mersenne numbers are defined by the following recurrence relation

[picture] | (2.1)

for with boundary conditions

Corollary 2.2. For , we get usual Mersenne sequence in [9] as follows

Theorem 2.3. The generating function for the k-order Mersenne numbers is

Proof. Let be the generating function of the k-order Mersenne numbers .

Since the recurrence relation for k-order Mersenne numbers is

the coefficients of are zero for . By taking parenthesis we get

Corollary 2.4. For we obtain the generating function of the Mersenne numbers as follows:

Theorem 2.5. The sum of the k-order Mersenne numbers is given by:

Proof. By the recurrence relation of the k-order Mersenne numbers (2.1) we have

From this equality

So we obtain

Adding and subtracting the following terms to above statement

we get

Consequently, we get

Corollary 2.6. For , we obtain the sum of the Mersenne numbers in [9] as follows:

Now we introduce the matrices and that plays the role of the Q-matrix for Fibonacci number. Let and determine the matrices defined as

and

Now we can give the following lemma and theorem.

Lemma 2.7. For we obtain as

Theorem 2.8. Let Then we get as follow:

[picture] | (2.2)

Proof. We can proof by induction method on . If then from the definition of the matrix and k-order Mersenne numbers

Assume that the theorem holds for

Then for we get

Theorem 2.9. Let be integer. Then

Theorem 2.10. Let be integer. Then

[picture] | (2.3)

Theorem 2.11. For any positive integer and

Proof. For ,

In this case, is obtained from (2.3) as follows:

If we use (2.2), we get as follows

and

Then we can obtain

and

If the equality is used, we get as

Corollary 2.12. For , we obtain for usual Mersenne numbers as follows

Example 2.13. For and , we get as

3. Conclusion

In this study, we defined k-order Mersenne numbers with the initial conditions. We gave an interesting result for and obtained Mersenne numbers. Thus, we got k-order Mersenne numbers, which is a new generalization of the usual Mersenne numbers. We introduced generating function of k-order Mersenne numbers and defined the sum of the first terms of these numbers. As with the matrix representation of other number sequences, we defined the Q-matrix for k-order Mersenne numbers and gave interesting theorems of matrix representations.

Acknowledgements

This work is supported by the Scientific Research Project (BAP) 2020FEBE003, Pamukkale University, Denizli, Turkey.

References

[1]   | Mcllroy, M.D., “Number theory in computer graphics the unreasonable effectiveness of number theory”, Amer. Math. Soc., 105, 1992.
[2]   | Box, G. E., Jenkins, G. M., “Times series analysis, Forecasting and Control”; Holden Day; San Francisco; 1970.
[3]   | Catarino, P., “On some identities and generating functions for k-Pell numbers”, International Journal of Mathematics and Analysis, 7(38), 1877, 2013.
[4]   | Catarino, P., “On some identities and generating function for k-Fibonacci sequence”, International Journal of Contemporary Mathematical Sciences, 9(1), 37, 2014.
[5]   | El-Mikkawy, M., Sogabe, T., “A new family of k-Fibonacci numbers”, Applied Mathematics and Computation, 215(12), 4456, 2010.
[6]   | Falcon, S., “On the k-Lucas numbers”, International Journal of Contemporary Mathematical Sciences, 6(21), 1039, 2011.
[7]   | Koshy, T., “Fibonacci and Lucas numbers with applications”, John Wiley & Sons; 2001.
[8]   | Horadam, A.F.,” Jacobsthal representation numbers”, Fibonacci Quarterly, 34, 1996.
[9]   | Catarino, P., Campos, H., Vasco, P., “On the mersenne sequence”, Annales Mathe-maticae et Informaticae, 46, 2016.
[10]   | Shannon, A.G., Cook, C.K., “Generalized Fibonacci-Feinberg sequencesé, Adv. Stud. Contemp. Math.(kyungshang), 21(2), 171-179, 2011.
[11]   | Tekcan, A., Ozkoc, A., Gezer, B., Bizim, O., “Some relations involving the sums of Fibonacci numbers”, Proc. Jangjeon Math. Soc., 11(1), 1-12, 2008.
[12]   | Kim, T., Dolgy, D. V., Kim, D.S., Seo, J.J., “Concolved Fibonacci numbers and their applications”, Ars Combin., 135, 119-131, 2017.
[13]   | Koshy, T., Gao, Z., “Catalan numbers with Mersenne subscripts”, Mathematical Scientist, 38, 86, 2013.
[14]   | Uslu, K., Deniz, V., “Some identities of k-Mersenne numbers”, Advances and Applications in Discrete Mathematics, 18(4), 413, 2017.
[15]   | Szynal-Liana, A., Wloch, I., “On generalized mersenne hybrid numbers”, Annales Universitatis Mariae Curie-Sklodowska Lublin-Polonia; 2020.
[16]   | Chelgham, M., Boussayoud, A., “On the k-Mersenne--Lucas numbers”, Notes on Number Theory and Discrete Mathematics, 27(1), 7, 2021
[17]   | Tasci, D., Sevgi, E., “Some properties between Mersenne, Jacobsthal and Jacobsthal-Lucas Hybrid Numbers”, Chaos, Solitons and Fractals, 146, 2021.