A New Generalization of Pell-Lucas Numbers (Bi-Periodic Pell-Lucas Sequence)

Sukran Uygun, Hasan Karatas

Abstract


In this study, we bring into light, a new generalization of the Jacobsthal Lucas numbers, which shall also be called the bi-periodic Jacobsthal Lucas sequence as
\begin{align*}
Q_{n}=
\begin{cases}
2bQ_{n-1}+Q_{n-2},&\text{if} \ n \ \text{is even} \\
2aQ_{n-1}+Q_{n-2},&\text{if} \ n \ \text{is odd}%
\end{cases} \text{\ \ }n\geq 2,
\end{align*}
with initial conditions \(Q_{0}=2\), \(Q_{1}=a\). The Binet formula as well as the generating function for this sequence are given. The convergence
properties of the consecutive terms of this sequence are also examined after which the well known Cassini, Catalans and the D'Ocagne's identities as well as some related summation formulas are also given.


Keywords


Bi-periodic Pell sequence; Pell-Lucas sequence; Generating function; Binet formula

Full Text:

PDF

References


G. Bilgici, Two generalizations of Lucas sequence, Applied Mathematics and Computation 245 (2014), 526 – 538, DOI: 10.1016/j.amc.2014.07.111.

A. Coskun and N. Taskara, The matrix sequence in terms of bi-periodic Fi-bonacci numbers, arXiv: https://arxiv.org/abs/1603.07487v2 [math.NT], 4 April 2016,

A. Coskun, N. Yilmaz and N. Taskara, A note on the bi-periodic Fibonacci and Lucas matrix sequences, arXiv: https://arxiv.org/abs/1604.00766v1 [math.NT] 4 April 2016,

M. Edson and O. Yayenie, A new generalization of Fibonacci sequences and the extended Binet’s formula, INTEGERS Electron. J. Comb. Number Theor. 9 (2009), 639 – 654, DOI: 10.1515/INTEG.2009.051.

A. H. George, Some formulae for the Fibonacci sequence with generalization, Fibonacci Quart. 7 (1969), 113 – 130.

A. F. Horadam, Pell identities, Fibonacci Quarterly 9(3) (1971), 245 – 252.

S. P. Jun and K. H. Choi, Some properties of the generalized Fibonacci sequence by matrix methods, Korean J. Math 24(4) (2016), 681 – 691, DOI: 10.11568/kjm.2016.24.4.681.

T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley and Sons Inc., New York (2001).

S. P. Pethe and C. N. Phadte, A generalization of the Fibonacci sequence, in: Applications of Fibonacci Numbers G. E. Bergum, A. N. Philippou and A. F. Horadam (eds.), Springer, Dordrecht (1993), DOI: 10.1007/978-94-011-2058-6_46.

S. Uygun and E. Owusu, A new generalization of Jacobsthal numbers (bi-periodic Jacobsthal sequences), Journal of Mathematical Analysis 7(5) (2016), 28 – 39.

S. Uygun and E. Owusu, Matrix representation of bi-periodic Jacobsthal, arXiv: https://arxiv.org/abs/1702.00604v1 [math.CO] 2 February 2017,

O. Yayenie, A note on generalized Fibonacci sequence, Appl. Math. Comput. 217(12) (2011), 5603 – 5611, DOI: 10.1016/j.amc.2010.12.038.




DOI: http://dx.doi.org/10.26713%2Fcma.v10i3.1106

Refbacks

  • There are currently no refbacks.


eISSN 0975-8607; pISSN 0976-5905