On the Bihyperbolic \(k\)-Fibonacci Numbers

Sergio Falcón

doi:10.26713/cma.v15i2.2592

Authors

Sergio Falcón Departamento de Matematicas, Universidad de Las Palmas de Gran Canari, 35017-Las Palmas de Gran Canaria, Spain https://orcid.org/0000-0001-9917-3101 (unauthenticated)

DOI:

https://doi.org/10.26713/cma.v15i2.2592

Keywords:

\(k\)-Fibonacci numbers, Hiperbolic and bihyperbolic numbers, Convolution, Binet, Catalan and d’Ocagne identities, Generating function

Abstract

In this paper, we extend the concept of bihyperbolic numbers to the \(k\)-Fibonacci numbers. We will find the main formulas that affect these numbers, such as the recurrence formulas and the sum of the first \(n\) terms, as well as the identities of Binet, Catalan and d’Ocagne. We finish the study by finding the generating function of the bihyperbolic \(k\)-Fibonacci numbers.

519 281

Downloads

Download data is not yet available.

References

D. Bród, A. Szynal-Liana and I. Włoch, On some combinatorial properties of bihyperbolic numbers of the Fibonacci type, Mathematical Methods in the Applied Sciences 44(6) (2021), 4607 – 4615, DOI: 10.1002/mma.7054.

S. Falcón and Á. Plaza, On the Fibonacci k-numbers, Chaos, Solitons & Fractals 32(5) (2007), 1615 – 1624, DOI: 10.1016/j.chaos.2006.09.022.

S. Falcón and Á. Plaza, The k-Fibonacci sequence and the Pascal 2-triangle, Chaos, Solitons & Fractals 33(1) (2007), 38 – 49, DOI: 10.1016/j.chaos.2006.10.022.

V. E. Hoggat, Fibonacci and Lucas Numbers, Houghton Mifflin Company, Palo Alto, USA, iv + 92 pages (1979).

A. F. Horadam, A generalized Fibonacci sequence, The American Mathematical Monthly 68(5) (1961), 455 – 459, DOI: 10.1080/00029890.1961.11989696.

D. S. Kulyabov, A. V. Korolkova and M. N. Gevorkyan, Hyperbolic numbers as Einstein numbers, Journal of Physics: Conference Series 1557(1) (2020), 012027, DOI: 10.1088/1742-6596/1557/1/012027.

G. Sobczyk, The hyperbolic number plane, The College Mathematical Journal 26(4) (1995), 268 – 280, DOI: 10.2307/2687027.

S. Vajda, Fibonacci & Lucas Numbers, and the Golden Section: Theory and Applications, Ellis Horwood Limited, Chichester (1989).