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On the m-extension of the Fibonacci and Lucas p-numbers

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  • Kocer, E. Gokcen
  • Tuglu, Naim
  • Stakhov, Alexey

Abstract

In this article, we define the m-extension of the Fibonacci and Lucas p-numbers (p⩾0 is integer and m>0 is real number) from which, specifying p and m, classic Fibonacci and Lucas numbers (p=1, m=1), Pell and Pell–Lucas numbers (p=1, m=2), Fibonacci and Lucas p-numbers (m=1), Fibonacci m-numbers (p=1), Pell and Pell–Lucas p-numbers (m=2) are obtained. Afterwards, we obtain the continuous functions for the m-extension of the Fibonacci and Lucas p-numbers using the generalized Binet formulas. Also we introduce in the article a new class of mathematical constants – the Golden (p,m)-Proportions, which are a wide generalization of the classical golden mean, the golden p-proportions and the golden m-proportions. The article is of fundamental interest for theoretical physics where Fibonacci numbers and the golden mean are used widely.

Suggested Citation

  • Kocer, E. Gokcen & Tuglu, Naim & Stakhov, Alexey, 2009. "On the m-extension of the Fibonacci and Lucas p-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1890-1906.
    • Handle: RePEc:eee:chsofr:v:40:y:2009:i:4:p:1890-1906
    DOI: 10.1016/j.chaos.2007.09.071
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    References listed on IDEAS

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    1. Stakhov, Alexey & Rozin, Boris, 2006. "The continuous functions for the Fibonacci and Lucas p-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 1014-1025.
    2. El Naschie, M.S., 2007. "Feigenbaum scenario for turbulence and Cantorian E-infinity theory of high energy particle physics," Chaos, Solitons & Fractals, Elsevier, vol. 32(3), pages 911-915.
    3. Stakhov, Alexey & Rozin, Boris, 2005. "The Golden Shofar," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 677-684.
    4. El Naschie, M.S., 2006. "Elementary prerequisites for E-infinity," Chaos, Solitons & Fractals, Elsevier, vol. 30(3), pages 579-605.
    5. Falcón, Sergio & Plaza, Ángel, 2008. "The k-Fibonacci hyperbolic functions," Chaos, Solitons & Fractals, Elsevier, vol. 38(2), pages 409-420.
    6. El Naschie, M.S., 2007. "On the topological ground state of E-infinity spacetime and the super string connection," Chaos, Solitons & Fractals, Elsevier, vol. 32(2), pages 468-470.
    7. Stakhov, Alexey & Rozin, Boris, 2006. "Theory of Binet formulas for Fibonacci and Lucas p-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 27(5), pages 1162-1177.
    8. Stakhov, Alexey, 2006. "Fundamentals of a new kind of mathematics based on the Golden Section," Chaos, Solitons & Fractals, Elsevier, vol. 27(5), pages 1124-1146.
    9. El Naschie, M.S., 2007. "From symmetry to particles," Chaos, Solitons & Fractals, Elsevier, vol. 32(2), pages 427-430.
    10. El Naschie, M.S., 2007. "Hilbert space, Poincaré dodecahedron and golden mean transfiniteness," Chaos, Solitons & Fractals, Elsevier, vol. 31(4), pages 787-793.
    11. Falcón, Sergio & Plaza, Ángel, 2007. "On the Fibonacci k-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 32(5), pages 1615-1624.
    12. Stakhov, A.P., 2005. "The Generalized Principle of the Golden Section and its applications in mathematics, science, and engineering," Chaos, Solitons & Fractals, Elsevier, vol. 26(2), pages 263-289.
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    Cited by:

    1. E. Gokcen Kocer & Huriye Alsan, 2022. "Generalized Hybrid Fibonacci and Lucas p-numbers," Indian Journal of Pure and Applied Mathematics, Springer, vol. 53(4), pages 948-955, December.
    2. Ivana Matoušová & Pavel Trojovský, 2020. "On Coding by (2, q )-Distance Fibonacci Numbers," Mathematics, MDPI, vol. 8(11), pages 1-24, November.
    3. Fiorenza, Alberto & Vincenzi, Giovanni, 2011. "Limit of ratio of consecutive terms for general order-k linear homogeneous recurrences with constant coefficients," Chaos, Solitons & Fractals, Elsevier, vol. 44(1), pages 145-152.
    4. Esmaeili, M. & Gulliver, T.A. & Kakhbod, A., 2009. "The Golden mean, Fibonacci matrices and partial weakly super-increasing sources," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 435-440.
    5. Florek, Wojciech, 2018. "A class of generalized Tribonacci sequences applied to counting problems," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 809-821.
    6. Hatir, E. & Noiri, T., 2009. "On δ–β-continuous functions," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 205-211.
    7. Ilija Tanackov & Ivan Pavkov & Željko Stević, 2020. "The New New-Nacci Method for Calculating the Roots of a Univariate Polynomial and Solution of Quintic Equation in Radicals," Mathematics, MDPI, vol. 8(5), pages 1-18, May.

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