Binomial Transform of the Generalized \(k\)-Fibonacci Numbers

Sergio Falcón

Abstract


We recall the concept and some properties of the generalized \(k\)-Fibonacci numbers and then apply the binomial transform to these sequences. As consequence, we obtain new integer sequences related to the generalized \(k\)-Fibonacci numbers. Finally, we find the recurrence relation of these new sequences and the formulas for their sums.


Keywords


\(k\)-Fibonacci numbers; Generalization of the \(k\)-Fibonacci numbers; Generating function; Binomial transform

Full Text:

PDF

References


P. Barry, On integer-sequence-based constructions og generalized pascal triangles, Journal of Integer Sequences 9, Article 06.2.4 (2006).

A. Benjamin, The Lucas triangle recounted, Applications of Fibonacci Numbers, Vol. 11, W. Webb (ed.), Kluwer Academic Publishers (2008), URL: https://www.math.hmc.edu/~benjamin/papers/LucasTriangle.pdf.

S. Falcón, On the k-Lucas Numbers, Int. J. Contemp. Math. Sciences 6(21) (2011), 1039 – 1050.

S. Falcón, On the k-Lucas triangle and its relationship with the k-Lucas numbers, J. Math. Comput. Sci. 6(3) (2012), 425 – 434.

S. Falcón, Generalized (k, r)-Fibonacci numbers, General Mathematics Notes 25(2) (2014), 148 – 158.

S. Falcón and A. Plaza, Binomial transforms of the k-Fibonacci sequence, International Journal of Nonlinear Sciences & Numerical Simulation 10(11-12) (2009), 1527 – 1538, DOI: 10.1515/IJNSNS.2009.10.11-12.1527.

S. Falcón and A. Plaza, On the Fibonacci k-numbers, Chaos, Solitons & Fractals (2006), DOI: 10.1016/j.chaos.2006.09.022, DOI: 10.1016/j.chaos.2006.09.022.

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

S. Falcón and A. Plaza, On k-Fibonacci numbers of arithmetic indexes, Applied Mathematics and Computation 208 (2009), 180 – 185, DOI: 10.1016/j.amc.2008.11.031.

S. Falcón and A. Plaza, Binomial transforms of the k-Fibonacci sequence, International Journal of Nonlinear Science and Numerical Simulation 10(11-12) (2009), 1527 – 1538, DOI: 10.1515/IJNSNS.2009.10.11-12.1527.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison Wesley Publishing Co. (1998).

V. E. Hogat, Fibonacci and Lucas Numbers, Houghton-Mifflin, Palo Alto (1969).

A. F. Horadam, A generalized Fibonacci sequence, Math Mag. 68 (1961), 455 – 459, DOI: 10.1080/00029890.1961.11989696.

N. J. A. Sloane, The On-line Encyclopedia of Integer Sequences, pp. 1– 9, https://arxiv.org/abs/math/0312448 (2006).

N. Robbins, Vieta’s triangular array and a related family of polynomials, Internat. J. Mayth. & Math. Sci. 14(2) (1991), 239 – 244, DOI: 10.1155/S0161171291000261.

A. G. Shanon and A. F. Horadam, Generalized Fibonacci triples, Amer. Math. Monthly 80 (1973), 187 – 190, DOI: 10.1080/00029890.1973.11993253.

V. W. Spinadel, The family of metallic means, Vis. Math. 1(3), available from http://members.tripod.com/vismath/ (1999).

I. Wloch, U. Bednarz, D. Bród, A. Wloch and M. Wolowiecz-Musial, On a new type of distance Fibonacci numbers, Discrete Applied Mathematics 161 (2013), 2695 – 2701, DOI: 10.1016/j.dam.2013.05.029.




DOI: http://dx.doi.org/10.26713%2Fcma.v10i3.1221

Refbacks

  • There are currently no refbacks.


eISSN 0975-8607; pISSN 0976-5905