On Generalized Pell Numbers of Order r ≥ 2

E. V. Pereira Spreafico, M. Rachidi

Abstract


In this paper we investigate the generalized Pell numbers of order r ≥ 2 through the properties of their related fundamental system of generalized Pell numbers. That is, the generalized Pell number of order r ≥ 2; are expressed as a linear combination of a fundamental system of generalized Pell numbers. The properties of this fundamental system are examined and results can be established for generalized Pell numbers of order r ≥ 2. Some identities and combinatorial results are established. Moreover, the analytic study of the fundamental system of generalized Pell is provided. Furthermore, the generalized Pell Cassini identity type is provided.


Keywords


Generalized Pell fundamental system; Generalized Pell numbers; Combinatorial identities; Analytic representations; Generalized Cassini identity

Full Text:

PDF

References


P. Catarino and P. Vasco, “On some identities and generating functions for k-pell-lucas sequence,” Appl. Math. Sci., vol. 7, no. 98, pp. 4867–4873, 2013.

A. Daşdemir, “On generalized order−k modified pell and pell–lucas numbers in terms of fibonacci and lucas numbers,” Notes on Number Theory and Discrete Mathematics, vol. 26, no. 2, pp. 205–212, 2020.

A. Shannon and A. Horadam, “Generalized pell numbers and polynomials,” Howard F.T. (eds) Applications of Fibonacci Numbers, Springer, Dordrecht, pp. 213–224, 2004.

A. Shannon and C. Wong, “Some properties of generalized third order pell numbers,” Notes on Number Theory and Discrete Mathematics, vol. 14, no. 4, pp. 16–24, 2008.

F. R. V. Alves and P. M. M. C. Catarino, “Generalized fibonacci and k-pell matrix sequences: Another way of demonstrating their properties,” Notes on Number Theory and Discrete Mathematics, vol. 25, no. 4, pp. 110–122, 2019.

R. B. Taher and M. Rachidi, “On the matrix powers and exponential by r- generalized fibonacci sequences methods: the companion matrix case,” Linear Algebra and Its Applications, vol. 370, pp. 341–353, 2003.

W. Chen and J. Louck, “The combinatorial power of the companion matrix,” Linear Algebra and its Applications, vol. 232, pp. 261–278, 1996.

G. Philippou, “On the k-th order linear recurrence and some probability appli- cations,” Applications of Fibonacci Numbers Eds G.E. Bergum, A.N. Philippou and A.F. Horadam, Kluwer Academic Publishers, 1988.

R. B. Taher and M. Rachidi, “Solving some generalized vandermonde systems and inverse of their associate matrices via new approaches for the binet for- mula,” Applied Mathematics and Computation, vol. 290, pp. 267–280, 2016.

R. P. Stanley, “Enumerative combinatorics volume 1 second edition,” Cam- bridge studies in advanced mathematics, 2011.




DOI: https://doi.org/10.5540/tcam.2021.022.01.00125

Article Metrics

Metrics Loading ...

Metrics powered by PLOS ALM

Refbacks

  • There are currently no refbacks.


Trends in Computational and Applied Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)

 

Indexed in:

                       

         

 

Desenvolvido por:

Logomarca da Lepidus Tecnologia