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Non-Newtonian Pell and Pell-Lucas numbers

Year 2024, Volume: 13 Issue: 1, 22 - 35, 30.04.2024
https://doi.org/10.54187/jnrs.1447678

Abstract

In the present paper, we introduce a new type of Pell and Pell-Lucas numbers in terms of non-Newtonian calculus, which we call non-Newtonian Pell and non-Newtonian Pell-Lucas numbers, respectively. In non-Newtonian calculus, we study some significant identities and formulas for classical Pell and Pell-Lucas numbers. Therefore, we derive some relations with non-Newtonian Pell and Pell-Lucas numbers. Furthermore, we investigate some properties of non-Newtonian Pell and Pell-Lucas numbers, including Catalan-like identities, Cassini-like identities, Binet-like formulas, and generating functions.

References

  • M. Grossman, R. Katz, Non-Newtonian calculus, Lee Press, Pigeon Cove, Massachusetts, 1972.
  • M. Grossman, An introduction to non-Newtonian calculus, International Journal of Mathematical Education in Science and Technology 10 (4) (1979) 525-528.
  • M. Grossman, The first nonlinear system of differential and integral calculus, Mathco, Rockport, Massachusetts, 1979.
  • M. Grossman, Bigeometric calculus: A system with a scale-free derivate, Archimedes Foundation, Rockport, Massachusetts, 1983.
  • D. Aerts, M. Czachor, M. Kuna, Simple fractal calculus from fractal arithmetic, Reports on Mathematical Physics 81 (3) (2018) 359-372.
  • A. E. Bashirov, E. M. Kurpınar, A. Özyapıcı, Multiplicative calculus and its applications, Journal of Mathematical Analysis and Applications 337 (1) (2008) 36-48.
  • A. E. Bashirov, E. Mısırlı, Y. Tandoğdu, A. Özyapıcı, On modeling with multiplicative differential equations, Applied Mathematics-A Journal of Chinese Universities 26 (4) (2011) 425-438.
  • K. Boruah, B. Hazarika, $G$-Calculus, TWMS Journal of Applied and Engineering Mathematics 8 (1) (2018) 94-105.
  • D. Campbell, Multiplicative calculus and student projects, Problems, Resources, and Issues in Mathematics Undergraduate Studies 9 (4) (1999) 327-332.
  • A. F. Çakmak, F. Başar, Some new results on sequence spaces with respect to non-Newtonian calculus, Journal of Inequalities and Applications 2012 (2012) Article Number 228 17 pages.
  • A. F. Çakmak, F. Başar, Certain spaces of functions over the field of non-Newtonian complex numbers, Abstract and Applied Analysis 2014 (2014) Article ID 236124 12 pages.
  • C. Duyar, M. Erdoğan, On non-Newtonian real number series, IOSR Journal of Mathematics 12 (2016) 34-48.
  • L. Florack, H. van Assen, Multiplicative calculus in biomedical image analysis, Journal of Mathematical Imaging and Vision 42 (2012) 64-75.
  • J. Grossman, M. Grossman, R. Katz, The first systems of weighted differential and integral calculus, Archimedes Foundation, Rockport, Massachusetts, 1980.
  • J. Grossman, Meta-calculus: Differential and integral, Archimedes Foundation, Rockport, Massachusetts, 1981.
  • U. Kadak, H. Efe, The construction of Hilbert spaces over the non-Newtonian field, International Journal of Analysis 2014 (2014) Article ID 746059 10 pages.
  • U. Kadak, Y. Gürefe, A generalization on weighted means and convex functions with respect to the non-Newtonian calculus, International Journal of Analysis 2016 (2016) Article ID 5416751 9 pages.
  • A. Özyapıcı, B. Bilgehan, Finite product representation via multiplicative calculus and its applications to exponential signal processing, Numerical Algorithms 71 (2016) 475-489.
  • D. Stanley, A multiplicative calculus, Problems, Resources, and Issues in Mathematics Undergraduate Studies 9 (4) (1999) 310-326.
  • D. F. M. Torres, On a non-Newtonian calculus of variations, Axioms 10 (2021) Article Number 171 15 pages.
  • M. Ç. Yılmazer, E. Yılmaz, S. Göktaş, M. Et, Multiplicative Laplace transform in $q$-calculus, Filomat 37 (18) (2023) 5859-5872.
  • V. E. Hoggatt Jr., Fibonacci and Lucas numbers, Houghton Mifflin Company, Boston, 1969.
  • T. Koshy, Fibonacci and Lucas numbers with applications, John Wiley and Sons, New York, 2001.
  • S. Vajda, Fibonacci and Lucas numbers, and the golden section: Theory and applications, Ellis Horwood Limited, Chichester, 1989.
  • M. Bicknell, A primer on the Pell sequence and related sequence, The Fibonacci Quarterly 13 (4) (1975) 345-349.
  • A. F. Horadam, Pell identities, The Fibonacci Quarterly 9 (3) (1971) 245-252.
  • T. Koshy, Pell and Pell-Lucas numbers with applications, Springer, New York, 2014.
  • G. Bilgici, New generalizations of Fibonacci and Lucas sequences, Applied Mathematical Sciences 8 (29) (2014) 1429-1437.
  • P. Catarino, On some identities and generating functions for $k$-Pell numbers, International Journal of Mathematical Analysis 7 (38) (2013) 1877-1883.
  • H. Civciv, R. Türkmen, On the $(s,t)$-Fibonacci and Fibonacci matrix sequences, Ars Combinatoria 87 (2008) 161-173.
  • H. Civciv, R. Türkmen, Notes on the $(s,t)$-Lucas and Lucas matrix sequences, Ars Combinatoria 89 (2008) 271-285.
  • C. B. Çimen, A. İpek, On Pell quaternions and Pell-Lucas quaternions, Advances in Applied Clifford Algebras 26 (2016) 39-51.
  • S. Falcon, A. Plaza, On the Fibonacci $k$-numbers, Chaos Solitons Fractals 32 (2007) 1615-1624.
  • H. H. Güleç, N. Taşkara, On the $(s,t)$-Pell and $(s,t)$-Pell-Lucas sequences and their matrix representations, Applied Mathematics Letters 25 (10) (2012) 1554-1559.
  • S. Halıcı, On Fibonacci quaternions, Advances in Applied Clifford Algebras 22 (2012) 321-327.
  • A. F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, The American Mathematical Monthly 70 (1963) 289-291.
  • A. F. Horadam, J. M. Mahon, Pell and Pell-Lucas polynomials, Fibonacci Quarterly 23 (1) (1985) 7-20.
  • A. Szynal-Liana, I. Wloch, The Pell quaternions and the Pell octonions, Advances in Applied Clifford Algebras 26 (2016) 435-440.
  • A. Szynal-Liana, I. Wloch, On Pell and Pell-Lucas hybrid numbers, Commentationes Mathematicae 58 (2018) 11-17.
  • A. Szynal-Liana, I. Wloch, The Fibonacci hybrid numbers, Utilitas Mathematica 110 (2019) 3-10.
  • T. Yağmur, New approach to Pell and Pell-Lucas sequences, Kyungpook Mathematical Journal 59 (1) (2019) 23-34.
  • N. Değirmen, C. Duyar, A new perspective on Fibonacci and Lucas numbers, Filomat 37 (28) (2023) 9561-9574.
Year 2024, Volume: 13 Issue: 1, 22 - 35, 30.04.2024
https://doi.org/10.54187/jnrs.1447678

Abstract

References

  • M. Grossman, R. Katz, Non-Newtonian calculus, Lee Press, Pigeon Cove, Massachusetts, 1972.
  • M. Grossman, An introduction to non-Newtonian calculus, International Journal of Mathematical Education in Science and Technology 10 (4) (1979) 525-528.
  • M. Grossman, The first nonlinear system of differential and integral calculus, Mathco, Rockport, Massachusetts, 1979.
  • M. Grossman, Bigeometric calculus: A system with a scale-free derivate, Archimedes Foundation, Rockport, Massachusetts, 1983.
  • D. Aerts, M. Czachor, M. Kuna, Simple fractal calculus from fractal arithmetic, Reports on Mathematical Physics 81 (3) (2018) 359-372.
  • A. E. Bashirov, E. M. Kurpınar, A. Özyapıcı, Multiplicative calculus and its applications, Journal of Mathematical Analysis and Applications 337 (1) (2008) 36-48.
  • A. E. Bashirov, E. Mısırlı, Y. Tandoğdu, A. Özyapıcı, On modeling with multiplicative differential equations, Applied Mathematics-A Journal of Chinese Universities 26 (4) (2011) 425-438.
  • K. Boruah, B. Hazarika, $G$-Calculus, TWMS Journal of Applied and Engineering Mathematics 8 (1) (2018) 94-105.
  • D. Campbell, Multiplicative calculus and student projects, Problems, Resources, and Issues in Mathematics Undergraduate Studies 9 (4) (1999) 327-332.
  • A. F. Çakmak, F. Başar, Some new results on sequence spaces with respect to non-Newtonian calculus, Journal of Inequalities and Applications 2012 (2012) Article Number 228 17 pages.
  • A. F. Çakmak, F. Başar, Certain spaces of functions over the field of non-Newtonian complex numbers, Abstract and Applied Analysis 2014 (2014) Article ID 236124 12 pages.
  • C. Duyar, M. Erdoğan, On non-Newtonian real number series, IOSR Journal of Mathematics 12 (2016) 34-48.
  • L. Florack, H. van Assen, Multiplicative calculus in biomedical image analysis, Journal of Mathematical Imaging and Vision 42 (2012) 64-75.
  • J. Grossman, M. Grossman, R. Katz, The first systems of weighted differential and integral calculus, Archimedes Foundation, Rockport, Massachusetts, 1980.
  • J. Grossman, Meta-calculus: Differential and integral, Archimedes Foundation, Rockport, Massachusetts, 1981.
  • U. Kadak, H. Efe, The construction of Hilbert spaces over the non-Newtonian field, International Journal of Analysis 2014 (2014) Article ID 746059 10 pages.
  • U. Kadak, Y. Gürefe, A generalization on weighted means and convex functions with respect to the non-Newtonian calculus, International Journal of Analysis 2016 (2016) Article ID 5416751 9 pages.
  • A. Özyapıcı, B. Bilgehan, Finite product representation via multiplicative calculus and its applications to exponential signal processing, Numerical Algorithms 71 (2016) 475-489.
  • D. Stanley, A multiplicative calculus, Problems, Resources, and Issues in Mathematics Undergraduate Studies 9 (4) (1999) 310-326.
  • D. F. M. Torres, On a non-Newtonian calculus of variations, Axioms 10 (2021) Article Number 171 15 pages.
  • M. Ç. Yılmazer, E. Yılmaz, S. Göktaş, M. Et, Multiplicative Laplace transform in $q$-calculus, Filomat 37 (18) (2023) 5859-5872.
  • V. E. Hoggatt Jr., Fibonacci and Lucas numbers, Houghton Mifflin Company, Boston, 1969.
  • T. Koshy, Fibonacci and Lucas numbers with applications, John Wiley and Sons, New York, 2001.
  • S. Vajda, Fibonacci and Lucas numbers, and the golden section: Theory and applications, Ellis Horwood Limited, Chichester, 1989.
  • M. Bicknell, A primer on the Pell sequence and related sequence, The Fibonacci Quarterly 13 (4) (1975) 345-349.
  • A. F. Horadam, Pell identities, The Fibonacci Quarterly 9 (3) (1971) 245-252.
  • T. Koshy, Pell and Pell-Lucas numbers with applications, Springer, New York, 2014.
  • G. Bilgici, New generalizations of Fibonacci and Lucas sequences, Applied Mathematical Sciences 8 (29) (2014) 1429-1437.
  • P. Catarino, On some identities and generating functions for $k$-Pell numbers, International Journal of Mathematical Analysis 7 (38) (2013) 1877-1883.
  • H. Civciv, R. Türkmen, On the $(s,t)$-Fibonacci and Fibonacci matrix sequences, Ars Combinatoria 87 (2008) 161-173.
  • H. Civciv, R. Türkmen, Notes on the $(s,t)$-Lucas and Lucas matrix sequences, Ars Combinatoria 89 (2008) 271-285.
  • C. B. Çimen, A. İpek, On Pell quaternions and Pell-Lucas quaternions, Advances in Applied Clifford Algebras 26 (2016) 39-51.
  • S. Falcon, A. Plaza, On the Fibonacci $k$-numbers, Chaos Solitons Fractals 32 (2007) 1615-1624.
  • H. H. Güleç, N. Taşkara, On the $(s,t)$-Pell and $(s,t)$-Pell-Lucas sequences and their matrix representations, Applied Mathematics Letters 25 (10) (2012) 1554-1559.
  • S. Halıcı, On Fibonacci quaternions, Advances in Applied Clifford Algebras 22 (2012) 321-327.
  • A. F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, The American Mathematical Monthly 70 (1963) 289-291.
  • A. F. Horadam, J. M. Mahon, Pell and Pell-Lucas polynomials, Fibonacci Quarterly 23 (1) (1985) 7-20.
  • A. Szynal-Liana, I. Wloch, The Pell quaternions and the Pell octonions, Advances in Applied Clifford Algebras 26 (2016) 435-440.
  • A. Szynal-Liana, I. Wloch, On Pell and Pell-Lucas hybrid numbers, Commentationes Mathematicae 58 (2018) 11-17.
  • A. Szynal-Liana, I. Wloch, The Fibonacci hybrid numbers, Utilitas Mathematica 110 (2019) 3-10.
  • T. Yağmur, New approach to Pell and Pell-Lucas sequences, Kyungpook Mathematical Journal 59 (1) (2019) 23-34.
  • N. Değirmen, C. Duyar, A new perspective on Fibonacci and Lucas numbers, Filomat 37 (28) (2023) 9561-9574.
There are 42 citations in total.

Details

Primary Language English
Subjects Algebra and Number Theory
Journal Section Articles
Authors

Tülay Yağmur 0000-0002-6224-1921

Publication Date April 30, 2024
Submission Date March 5, 2024
Acceptance Date April 26, 2024
Published in Issue Year 2024 Volume: 13 Issue: 1

Cite

APA Yağmur, T. (2024). Non-Newtonian Pell and Pell-Lucas numbers. Journal of New Results in Science, 13(1), 22-35. https://doi.org/10.54187/jnrs.1447678
AMA Yağmur T. Non-Newtonian Pell and Pell-Lucas numbers. JNRS. April 2024;13(1):22-35. doi:10.54187/jnrs.1447678
Chicago Yağmur, Tülay. “Non-Newtonian Pell and Pell-Lucas Numbers”. Journal of New Results in Science 13, no. 1 (April 2024): 22-35. https://doi.org/10.54187/jnrs.1447678.
EndNote Yağmur T (April 1, 2024) Non-Newtonian Pell and Pell-Lucas numbers. Journal of New Results in Science 13 1 22–35.
IEEE T. Yağmur, “Non-Newtonian Pell and Pell-Lucas numbers”, JNRS, vol. 13, no. 1, pp. 22–35, 2024, doi: 10.54187/jnrs.1447678.
ISNAD Yağmur, Tülay. “Non-Newtonian Pell and Pell-Lucas Numbers”. Journal of New Results in Science 13/1 (April 2024), 22-35. https://doi.org/10.54187/jnrs.1447678.
JAMA Yağmur T. Non-Newtonian Pell and Pell-Lucas numbers. JNRS. 2024;13:22–35.
MLA Yağmur, Tülay. “Non-Newtonian Pell and Pell-Lucas Numbers”. Journal of New Results in Science, vol. 13, no. 1, 2024, pp. 22-35, doi:10.54187/jnrs.1447678.
Vancouver Yağmur T. Non-Newtonian Pell and Pell-Lucas numbers. JNRS. 2024;13(1):22-35.


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