The ?th Power of Generalized (?, ?)-Jacobsthal and (?, ?)-Jacobsthal Lucas Matrix Sequences and Some Combinatorial Properties

The ?th Power of Generalized (?, ?)-Jacobsthal and (?, ?)-Jacobsthal Lucas Matrix Sequences and Some Combinatorial Properties

In this study, new formulas for the ?th power of (?, ?)-Jacobsthal and (?, ?)-Jacobsthal Lucas special matrix sequences are established by using determinant and trace of the matrices. By these formulas, some identities for (?, ?)-Jacobsthal and (?, ?)-Jacobsthal Lucas sequences are obtained. The formulas for finding the ?th power for classic Jacobsthal and Jacobsthal Lucas matrix sequences are also derivable if we choose ? = ? = 1.

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  • [1] K. S. Williams, The ?th Power of a 2 × 2 Matrix, Mathematics Magazine 65(5) (1992) 336-336.
  • [2] J. Mc Laughlin, Combinatorial Identities Deriving from the ?th Power of a 2 × 2 Matrix, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004) 1-15.
  • [3] J. Mc Laughlin, B. Sury, Powers of Matrix and Combinatorial Identities, Integers: Electronic Journal of Combinatorial Number Theory 5 (2005) 1-9.
  • [4] H. Belbachir, Linear Recurrent Sequences and Powers of a Square Matrix, Integers: Electronic Journal of Combinatorial Number Theory 6 (2006) 1-17.
  • [5] G. E. Bergum, V. E. Hoggatt Jr., Sums and products for recurring sequences, The Fibonacci Quarterly, 13(2) (1975) 115-120.
  • [6] Z. Akyüz, S. Halıcı, On Some Combinatorial Identities Involving the Terms of Generalized Fibonacci and Lucas Sequences, Hacettepe Journal of Mathematics and Statistics 42(4) (2013) 431-435.
  • [7] Z. Akyüz, S. Halıcı, Some Identities Deriving from the ?th Power of a Special Matrix. Advances in Difference Equations 1 (2012) 1-6.
  • [8] S. Uygun, The (?, ?)-Jacobsthal and (?, ?)-Jacobsthal Lucas Sequences, Applied Mathematical Sciences 70(9) (2015) 3467-3476.
  • [9] K. Uslu, S. Uygun, The (?, ?)-Jacobsthal and (?, ?)-Jacobsthal-Lucas Matrix Sequences, ARS Combinatoria 108 (2013) 13-22.
  • [10] S. Uygun, Some Sum Formulas of (?, ?)-Jacobsthal and (?, ?)-Jacobsthal Lucas Matrix Sequences, Applied Mathematics 7 (2016) 61-69.
  • [11] S. Halıcı, M. Uysal, A Study on Some identities involving (?? , ?)-Jacobsthal Numbers, Notes on Number Theory and Discrete Mathematics 26(4) (2020) 4 74-79. DOI: 10.7546/nntdm.2020.26.4.74-79
  • [12] A. A. Wani, P. Catarino, S. Halıcı, On a Study of (?, ?)-generalized Pell Sequence and Its Matrix Sequence, Journal of Mathematics 51(9) (2019) 17-32.
  • [13] A. F. Horadam, Jacobsthal Representation Numbers, The Fibonacci Quarterly 34(1) (1996) 40-54.