Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

New Approach to Pell and Pell-Lucas Sequences

  • Received : 2017.07.26
  • Accepted : 2018.12.06
  • Published : 2019.03.23
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we first define generalizations of Pell and Pell-Lucas sequences by the recurrence relations $$p_n=2ap_{n-1}+(b-a^2)p_{n-2}\;and\;q_n=2aq_{n-1}+(b-a^2)q_{n-2}$$ with initial conditions $p_0=0$, $p_1=1$, and $p_0=2$, $p_1=2a$, respectively. We give generating functions and Binet's formulas for these sequences. Also, we obtain some identities of these sequences.

Keywords

References

  1. G. Bilgici, New generalizations of Fibonacci and Lucas sequences, Appl. Math. Sc., 8(29)(2014), 1429-1437.
  2. G. Bilgici, Two generalizations of Lucas sequence, Appl. Math. Comput., 245(2014), 526-538. https://doi.org/10.1016/j.amc.2014.07.111
  3. A. Dasdemir, On the Pell, Pell-Lucas and modified Pell numbers by matrix method, Appl. Math. Sci., 5(64)(2011), 3173-3181.
  4. M. Edson and O. Yayenie, A new generalization of Fibonacci sequence and extended Binet's formula, Integers, 9(2009), 639-654. https://doi.org/10.1515/INTEG.2009.051
  5. H. H. Gulec and N. Taskara, On the (s, t)-Pell and (s, t)-Pell-Lucas sequences and their matrix represantations, Appl. Math. Lett., 25(2012), 1554-1559. https://doi.org/10.1016/j.aml.2012.01.014
  6. A. F. Horadam, A generalized Fibonacci sequence, Amer. Math. Monthly, 68(1961), 455-459. https://doi.org/10.1080/00029890.1961.11989696
  7. A. F. Horadam, Pell identities, Fibonacci Quart., 9(3)(1971), 245-263.
  8. T. Koshy, Pell and Pell-Lucas numbers with applications, Springer, Berlin, 2014.
  9. S. P. Pethe and C. N. Phadte, A generalization of the Fibonacci sequence, Applications of Fibonacci numbers, 5(St. Andrews, 1992), 465-472.