Abstract
Bu araştırmada, Gaussian genelleştirilmiş Leonardo sayılarını tanıtıyor ve kapsamlı bir şekilde inceliyoruz ve üç farklı duruma odaklanıyoruz: Gaussian modifiye Leonardo sayıları, Gaussian Leonardo-Lucas sayıları ve Gaussian Leonardo sayıları. Amacımız, bu dizilerin davranışı ve özellikleri hakkında kapsamlı bir anlayış sunmaktır.
Bu amaçla, bu dizilerle ilişkili çeşitli özdeşlikler ve matrisler türeterek derinlemesine bir analiz gerçekleştiriyoruz. Ayrıca, yineleme bağıntıları, Binet formülleri, üreteç fonksiyonlar, Simpson formülü, Honsberger özdeşliği ve çeşitli toplam formülleri gibi temel matematiksel araçları da araştırıyoruz. Bu çok yönlü yaklaşım, bu Gaussian tabanlı dizilerin yapısı ve davranışı hakkında değerli içgörüler sağlar. Sunduğumuz sonuçlar yalnızca mevcut bilgiyi genişletmekle kalmıyor, aynı zamanda Gaussian genelleştirilmiş Leonardo sayılarının daha fazla genelleştirilmesini ve uygulamasını araştırabilecek gelecekteki çalışmalar için de kapı açıyor
Keywords
Gaussian genelleştirilmiş Leonardo sayıları Gaussian Leonardo-Lucas sayıları Gaussian modifiye Leonardo sayıları Gaussian Leonardo sayıları
Ethical Statement
Bu çalışmanın, özgün bir çalışma olduğunu; çalışmanın hazırlık, veri toplama, analiz ve bilgilerin sunumu olmak üzere tüm aşamalarından bilimsel etik ilke ve kurallarına uygun davrandığımı; bu çalışma kapsamında elde edilmeyen tüm veri ve bilgiler için kaynak gösterdiğimi ve bu kaynaklara kaynakçada yer verdiğimi; kullanılan verilerde herhangi bir değişiklik yapmadığımı, çalışmanın Committee on Publication Ethics (COPE)' in tüm şartlarını ve koşullarını kabul ederek etik görev ve sorumluluklara riayet ettiğimi beyan ederim. Herhangi bir zamanda, çalışmayla ilgili yaptığım bu beyana aykırı bir durumun saptanması durumunda, ortaya çıkacak tüm ahlaki ve hukuki sonuçlara razı olduğumu bildiririm.
Supporting Institution
Yok
Thanks
Karaelmas Fen ve Mühendislik dergisi editör ve hakemlerine makalenin incelenmesi ve yayınlanması aşamasında gösterecekleri katkı ve yardımlarından ötürü teşekkür ederim.
References
- Alp, Y., Kocer, EG. 2021. Some properties of Leonardo numbers. Konuralp J. Math; 9 (1): 183–189.
- Aşçı, M., Gürel, E. 2013. Gaussian Jacobsthal and Gaussian Jacobsthal Lucas numbers. Ars Combinatoria, 111:53-62.
- Bednarz, U., Wołowiec-Musiał, M. 2023. Generalized Fibonacci–Leonardo numbers. Journal of Difference Equations and Applications, 30 (1): 111–121. DOI:10.1080/10236198.2023.2265509
- Catarino P., Borges A. 2020b. A note on incomplete Leonardo numbers. Integers: Electronic Journal of Combinatorial Number Theory, 20.
- Catarino, P., Borges A. 2020a. On Leonardo numbers. Acta Mathematica Universitatis Comenianae, 89 (1):75–86.
- Cerda‐Morales, G. 2022. On Gauss third‐order Jacobsthal numbers and their applications. Annals of the Alexandru Ioan Cuza University‐Mathematics, 67(2):231‐241.
- Frontczak, R. 2018. Convolutions for Generalized Tribonacci Numbers and Related Results. International Journal of Mathematical Analysis, 12(7):307 ‐ 324.
- Halıcı, S., Öz, S. 2016. On Some Gaussian Pell And Pell‐Lucas Numbers, Ordu Univ. J. Sci. Tech., 6(1):8‐18.
- Horadam, AF. 1984. Complex Fibonacci Numbers and Fibonacci Quaternions. The American Mathematical Monthly, 70(3):289‐291.
- Karataş, A. 2022. Complex Leonardo numbers, Notes on Number Theory and Discrete Mathematics, 28(3):458-465. DOI: 10.7546/nntdm.2022.28.3.458-465
- Pethe, S., Horadam, AF. 1986. Gaussian Generalized Fibonacci Numbers. Bulletin of the Australian Mathematical Society, 33(1):37‐48.
- Shannon, AG. 2019. A note on generalized Leonardo numbers. Notes Number Theory Discrete Mathematics, 25(3): 97–101. DOI: 10.7546/nntdm.2019.25.3.97-101
- Shannon, AG., Deveci, Ö. 2022. A note on generalized and extended Leonardo sequences. Notes on Number Theory and Discrete Mathematics, 28(1), 109-114, DOI: 10.7546/nntdm.2022.28.1.109-114. DOI: 10.7546/nntdm.2022.28.1.109-114
- Sloane, NJA. 1964. The on‐line encyclopedia of integer sequences, http://oeis.org/
- Soykan, Y. 2019. Simson Identity of Generalized m‐step Fibonacci Numbers. Int. J. Adv. Appl. Math. and Mech. 7(2):45‐56.
- Soykan, Y. 2020. A Study On Generalized (r,s,t)‐Numbers. MathLAB Journal, 7:101‐129.
- Soykan, Y. 2021. Generalized Leonardo numbers. Journal of Progressive Research in Mathematics, 18(4):58‐84.
- Soykan, Y. 2022. Generalized Guglielmo Numbers: An Investigation of Properties of Triangular, Oblong and Pentagonal Numbers via Their Third Order Linear Recurrence Relations. Earthline Journal of Mathematical Sciences, 9:1‐39.
- Soykan, Y. 2023. Generalized Tribonacci polynomials. Earthline Journal of Mathematical Sciences, 13 (1):1‐120.
- Soykan, Y. Taşdemir, E., Okumuş, İ. Göcen, M. 2018. Gaussian Generalized Tribonacci Numbers. Journal of Progressive Research in Mathematics, 14(2):2373-2387.
- Taşçı, D. 2018a. Gaussian Padovan and Gaussian Pell‐ Padovan numbers. Commun. Fac. Sci. Univ. Ank. Ser. Al Math. Stat., 67 (2):82‐88.
- Taşçı, D. 2018b. Gaussian Balancing and Gaussian Lucas Balancing Numbers. Journal of Science and Arts, 3(44):661‐666.
- Taşçı, D. 2021. On Mersenne Numbers. Journal of Science and Arts, 4 (57):1021‐1028.
- Tan, E., Leung H. 2023. On Leonardo p-numbers. Integers: Electronic Journal of Combinatorial Number Theory, 23(7). Doi: 10.5281/zenodo.7569221.
- Vieira, RPM., Alves, FRV., Catarino, PM. 2019, Relacoes bidimensionais e identidades da sequencia de leonardo. Revista Sergipana de Matematica e Educacao Matemat- ica; 2:156–73. Doi: 10.34179/revisem.v4i2.11863 .
- Yılmaz, F., Ertaş, A. 2023. On Quaternions with Gaussian Oresme Coefficients. Turkish Journal of Mathematics and Computer Science, 15(1):192‐202.
Abstract
In this research, we introduce and thoroughly examine Gaussian generalized Leonardo numbers, focusing on three distinct cases: Gaussian modified Leonardo numbers, Gaussian Leonardo‐Lucas numbers, and Gaussian Leonardo numbers. Our aim is to offer a comprehensive understanding of the behaviour and properties of these sequences.
To this end, we perform a detailed analysis, deriving various identities and matrices associated with these sequences. We also explore key mathematical tools such as recurrence relations, Binet’s formulas, generating functions, Simpson’s formula, Honsberger’s identity, and several summation formulas. This multifaceted approach provides valuable insights into the structure and behaviour of these Gaussian-based sequences. The results we present not only extend existing knowledge but also open the door for future studies that could explore further generalizations and applications of Gaussian generalized Leonardo numbers.
Keywords
Gaussian generalized Leonardo numbers Gaussian Leonardo‐Lucas numbers Gaussian modified Leonardo numbers Gaussian Leonardo numbers.
Ethical Statement
I declare that this study is an original study; that I have acted in accordance with scientific ethical principles and rules in all stages of the study, including preparation, data collection, analysis and presentation of information; that I have cited sources for all data and information not obtained within the scope of this study and included these sources in the bibliography; that I have not made any changes to the data used, that I have accepted all terms and conditions of the Committee on Publication Ethics (COPE) and have complied with ethical duties and responsibilities. I declare that I accept all moral and legal consequences that will arise in the event that a situation contrary to this declaration I have made regarding the study is detected at any time.
Supporting Institution
Yok
Thanks
I would like to thank the editors and referees of Karaelmas Science and Engineering Journal for their contributions and assistance during the review and publication of the article.
References
- Alp, Y., Kocer, EG. 2021. Some properties of Leonardo numbers. Konuralp J. Math; 9 (1): 183–189.
- Aşçı, M., Gürel, E. 2013. Gaussian Jacobsthal and Gaussian Jacobsthal Lucas numbers. Ars Combinatoria, 111:53-62.
- Bednarz, U., Wołowiec-Musiał, M. 2023. Generalized Fibonacci–Leonardo numbers. Journal of Difference Equations and Applications, 30 (1): 111–121. DOI:10.1080/10236198.2023.2265509
- Catarino P., Borges A. 2020b. A note on incomplete Leonardo numbers. Integers: Electronic Journal of Combinatorial Number Theory, 20.
- Catarino, P., Borges A. 2020a. On Leonardo numbers. Acta Mathematica Universitatis Comenianae, 89 (1):75–86.
- Cerda‐Morales, G. 2022. On Gauss third‐order Jacobsthal numbers and their applications. Annals of the Alexandru Ioan Cuza University‐Mathematics, 67(2):231‐241.
- Frontczak, R. 2018. Convolutions for Generalized Tribonacci Numbers and Related Results. International Journal of Mathematical Analysis, 12(7):307 ‐ 324.
- Halıcı, S., Öz, S. 2016. On Some Gaussian Pell And Pell‐Lucas Numbers, Ordu Univ. J. Sci. Tech., 6(1):8‐18.
- Horadam, AF. 1984. Complex Fibonacci Numbers and Fibonacci Quaternions. The American Mathematical Monthly, 70(3):289‐291.
- Karataş, A. 2022. Complex Leonardo numbers, Notes on Number Theory and Discrete Mathematics, 28(3):458-465. DOI: 10.7546/nntdm.2022.28.3.458-465
- Pethe, S., Horadam, AF. 1986. Gaussian Generalized Fibonacci Numbers. Bulletin of the Australian Mathematical Society, 33(1):37‐48.
- Shannon, AG. 2019. A note on generalized Leonardo numbers. Notes Number Theory Discrete Mathematics, 25(3): 97–101. DOI: 10.7546/nntdm.2019.25.3.97-101
- Shannon, AG., Deveci, Ö. 2022. A note on generalized and extended Leonardo sequences. Notes on Number Theory and Discrete Mathematics, 28(1), 109-114, DOI: 10.7546/nntdm.2022.28.1.109-114. DOI: 10.7546/nntdm.2022.28.1.109-114
- Sloane, NJA. 1964. The on‐line encyclopedia of integer sequences, http://oeis.org/
- Soykan, Y. 2019. Simson Identity of Generalized m‐step Fibonacci Numbers. Int. J. Adv. Appl. Math. and Mech. 7(2):45‐56.
- Soykan, Y. 2020. A Study On Generalized (r,s,t)‐Numbers. MathLAB Journal, 7:101‐129.
- Soykan, Y. 2021. Generalized Leonardo numbers. Journal of Progressive Research in Mathematics, 18(4):58‐84.
- Soykan, Y. 2022. Generalized Guglielmo Numbers: An Investigation of Properties of Triangular, Oblong and Pentagonal Numbers via Their Third Order Linear Recurrence Relations. Earthline Journal of Mathematical Sciences, 9:1‐39.
- Soykan, Y. 2023. Generalized Tribonacci polynomials. Earthline Journal of Mathematical Sciences, 13 (1):1‐120.
- Soykan, Y. Taşdemir, E., Okumuş, İ. Göcen, M. 2018. Gaussian Generalized Tribonacci Numbers. Journal of Progressive Research in Mathematics, 14(2):2373-2387.
- Taşçı, D. 2018a. Gaussian Padovan and Gaussian Pell‐ Padovan numbers. Commun. Fac. Sci. Univ. Ank. Ser. Al Math. Stat., 67 (2):82‐88.
- Taşçı, D. 2018b. Gaussian Balancing and Gaussian Lucas Balancing Numbers. Journal of Science and Arts, 3(44):661‐666.
- Taşçı, D. 2021. On Mersenne Numbers. Journal of Science and Arts, 4 (57):1021‐1028.
- Tan, E., Leung H. 2023. On Leonardo p-numbers. Integers: Electronic Journal of Combinatorial Number Theory, 23(7). Doi: 10.5281/zenodo.7569221.
- Vieira, RPM., Alves, FRV., Catarino, PM. 2019, Relacoes bidimensionais e identidades da sequencia de leonardo. Revista Sergipana de Matematica e Educacao Matemat- ica; 2:156–73. Doi: 10.34179/revisem.v4i2.11863 .
- Yılmaz, F., Ertaş, A. 2023. On Quaternions with Gaussian Oresme Coefficients. Turkish Journal of Mathematics and Computer Science, 15(1):192‐202.
Details
| Primary Language | English |
|---|---|
| Subjects | Algebra and Number Theory |
| Journal Section | Research Article |
| Authors | |
| Publication Date | April 22, 2025 |
| Submission Date | November 19, 2024 |
| Acceptance Date | February 13, 2025 |
| Published in Issue | Year 2025 Volume: 15 Issue: 1 |
