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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.
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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
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Journal of Cancer Research and Treatment
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Label_Title

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

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Label_Title

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

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Some Results on Gaussian Leonardo P Numbers

Mustafa Asci, Mustafa Yilmaz*

Pamukkale University, Department of Mathematics, Denizli, Turkey

Abstract: In this paper, we define the Gaussian Leonardo p numbers and we examine sum formula of the Gaussian Leonardo p numbers and then we define Q matrix of the Gaussian Leonardo p numbers. Also we give Binet formula of these numbers. We give the Gaussian Leonardo p numbers relations with the Leonardo p numbers and the Fibonacci p numbers.

Keywords: Leonardo numbers, Gaussian Leonardo numbers, Leonardo p numbers

Received February 18, 2025; Revised March 19, 2025; Accepted March 26, 2025

Citation: Mustafa Asci, Mustafa Yilmaz, Some Results on Gaussian Leonardo P Numbers. Turkish Journal of Analysis and Number Theory, 13 (1). 1-4. January 2025. doi:10.12691/tjant-13-1-1

1. Introduction

The Fibonacci sequence, a well-known example of an integer sequence, finds applications in mathematics computer science, and art. The Fibonacci and Lucas sequences are widely recognized as being among the most prominent integer sequences. In the realm complex numbers, Gaussian integers are characterized by having integer values for both their real and imaginary components. Gaussian integers, with their captivating properties, have drawn the attention of numerous researchers throughout history. Gauss’s groundbreaking contributions mathematics included the introduction of Gaussian integers in [1]. Complex Fibonacci numbers, introduced by Horadam, have found applications in various fields including number theory, algebra, and physics in [2]. The concepts of Gaussian Fibonacci and Gaussian Lucas numbers were brought forth by J.H. Jordan’s innovative research in [3] and he expanded the applicability of established relationships between Fibonacci sequences to Gaussian Fibonacci numbers. Pethe and Horadam introduced generalized form of Gaussian Fibonacci numbers, expanding the mathematical landscape in [4]. Berzsenyi’s work marked a significant milestone in the extension of Fibonacci numbers to the complex plane [5]. Furthermore, Stakhov and Rozin derived explicit Binet formulas for Fibonacci and Lucas p-numbers, extending the classical Binet formulas in [6]. Tasci studied in [7,8,9,10] complex Fibonacci p numbers, Gaussian Padovan and Gaussian Pell-Padovan sequences, Gaussian Mersenne numbers, Gauss balancing numbers, and Gauss Lucas-balancing numbers.

Fibonacci and Lucas numbers are defined recursively for n≥1:

, with the initial conditions and , with the initial conditions , respectively see [11]. Gaussian Fibonacci numbers are also [3] defined recursively: with the initial values

. It is clear that these numbers are closely related to Fibonacci numbers:, where . Similarly Gaussian Lucas numbers are defined [3] recursively for : with the initial conditions . Leonardo numbers are studied by Catarino and Burgers in [12,13,14]. They defined these numbers by the second order inhomogeneous recurrence relation :, with the initial conditions . Also, these numbers can be defined as : . Gaussian Leonardo numbers are studied by D.Tasci in [15]. Gaussian Leonardo numbers are defined as , with the initial conditions . Some Gaussian Leonardo numbers: It is clear that Gaussian Leonardo numbers are closely related to Leonardo numbers: where denotes the nth Leonardo number. Also for it holds that .There are many important generalizations of Fibonacci numbers. The generalized Fibonacci p- numbers [16] examples of them and are defined by:

with the initial conditions The generalized Lucas p-numbers are defined by : with the initial conditions and where p=0,1,2 and . Let p be an integer the Gaussian Fibonacci p-numbers are defined [17] by the following recurrence relation

with the initial conditions and . It can be easily seen that where is the nth Fibonacci p-number. For any given integer p>0, the Leonardo p-sequence is defined [18] by the following non-homogenous relation :

with the initial values . The non-homogenous recurrence relation of Leonardo p numbers can be converted to the following homogenous recurrence relation :

Now we will define the Gaussian Leonardo p-numbers, we will give some results of these numbers.

2. Main Results

We start by giving the definition of the Gaussian Leonardo p-sequence.

Definition 2.1

For any given integer p >0, the Gaussian Leoanardo p –sequence is defined by the following non-homogenous relation :

with the initial values and .

It is clear that Gaussian Leonardo p-numbers are closely related to Leoanrdo p-numbers:

The non-homogenous recurrence relation of Gaussian Leonardo p-numbers can be converted to the following homogenous recurrence relation :

For p=1, we get Lemma 2.1 in [17]

For p=1, p=2 and p=3 we give some of the numbers as follows

Theorem 2.1

For ,we have

where denotes the n-th Fibonacci p-numbers.

Proof 2.1

For , we have

and then we can get

Corollary 2.1

For p =1 , we get Theorem 2.1 in [17]

Theorem 2.2

For , we have

where and denotes respectively the n-th Gaussian Leonardo p –numbers,Lucas p-numbers and Fibonacci p-numbers.

Proof 2.2

For ,we have

and then

we can get

Corollary 2.2

For p=1, we get theorem 2.2.b in [17]

Theorem 2.3

The Binet formula of the Gaussian Leonardo p–sequence is

Corollary 2.3

For p=1, we get theorem 2.4 in [15]

Theorem 2.4

For , we have

Proof 2.4

We have ,

and

Corallary 2.4

For p=1, we get theorem 2.6 in [15]

Now we introduce the matrices that play the role of the Q-matrix.

Theorem 2.5

Let .Then

Proof 2.5

Theorem can be proved by mathematical induction on n.

Conclusion

This paper explores the properties and applications of Gaussian Leonardo p- numbers. Gaussian Leonardo p-numbers can be extended to Gaussian Lenardo p-polynomials.

Conflicts of Interest

The authors declare no conflict of interest.

ACKNOWLEDGEMENTS

The authors thank to the anonymous referees for his/her comments and valuable suggestions that improved the presentation of the manuscript.

References

[1]   | C. Gauss, Theoria residuorum biquadraticorum, Commentario Prima (1828) [Werke, Vol. II, pp. 65--92].
[2]   | A. F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Monthly 70 (1969) 289—291.
[3]   | J. H. Jordan, Gaussian Fibonacci and Lucas numbers, Fibonacci Quart. 3 (1965) 315--318.
[4]   | S. Pethe, A. F. Horadam, Generalized Gaussian Fibonacci numbers, Bull. Aust. Math. Soc. 33 (1986) 37--48.
[5]   | G. Berzsenyi, Gaussian Fibonacci numbers, Fibonacci Quart. 15 (1977) 233--236.
[6]   | Stakhov AP, Rozin B. "Theory of Binet formulas for Fibonacci and Lucas p-numbers". Chaos, Solitions & Fractals 2006; 27: 1162-77.
[7]   | D. Tasci, Gaussian Balancing numbers and Gaussian Lucas-Balancing numbers, J. Sci. Arts 44 (2018) 661--666.
[8]   | D. Tasci, Gaussian Padovan and Gaussian Pell-Padovan sequences, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat. 67 (2018) 82--88.
[9]   | D. Tasci, On Gaussian Mersenne numbers, J. Sci. Arts 57 (2021) 1021--1028.
[10]   | D. Tasci, F. Yalcin, Complex Fibonacci p-numbers, Commun. Math. Appl. 4 (2013) 213--218.
[11]   | Koshy T. "Fibonacci and Lucas Numbers with Applications," A Wiley-Interscience Publication, (2001)
[12]   | P. Catarino, A. Borges, A note on Gaussian modified Pell numbers, J. Inf. Optim. Sci. 39 (2018) 1363—1371.
[13]   | P. Catarino, A. Borges, A note on incomplete Leonardo numbers, Integers 20 (2020) #A43.
[14]   | P. Catarino, A. Borges, On Leonardo numbers, Acta Math. Univ. Comenian. 89 (2020) 75--86.
[15]   | D. Tasci On Gaussian Leonardo numbers. Contrib. Math. 2023, 7, 34--40.
[16]   | Stakhov AP. "Introduction into algorithmic measurement theory". Soviet Radio, Moscow (1977) [in Russian].
[17]   | M. Asci, E. Gurel "Gaussian Fibonacci and Gaussian Lucas p-Numbers" Ars Combin. 132, (2017), 389-402.
[18]   | Tan, E.; Leung, H.H. On Leonardo p-numbers. Integers 2023, 23, 1--11.