Some identities on conditional sequences by using matrix method

Tan, Elif; Ekin, Ali Bülent

doi:10.18514/mmn.2017.1321

Cited by 11 publications

(8 citation statements)

References 4 publications

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“…For the case of w 0 = 0, w 1 = 1 and c = 1, the sequence {w n } reduces to the bi-periodic Fibonacci sequence, and some basic properties of this sequence can be found in [4,10,18]. Its companion sequence, the bi-periodic Lucas sequence, was studied in [2,6,14,15]. For the case of c = 1, the sequence {w n } reduces to the bi-periodic Horadam sequence, and several properties of this sequence were given in [4,13].…”

Section: Introductionmentioning

confidence: 99%

“…For the detailed history of the matrix technique see [3,7,8,11,16,17]. The 2 × 2 matrix representation for the general case of the sequence {w n } was given firstly in [15], and several properties were obtained for the even indices terms of this sequence. Then, in [12], the author defined a new matrix identity for the bi-periodic Fibonacci sequence as follows:…”

Section: Introductionmentioning

confidence: 99%

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Some basic properties of the generalized bi-periodic Fibonacci and Lucas sequences

Tan

Leung

2020

Adv Differ Equ

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In this paper, we consider a generalization of Horadam sequence {w n } which is defined by the recurrence relation w n = χ (n)w n-1 + cw n-2 , where χ (n) = a if n is even, χ (n) = b if n is odd with arbitrary initial conditions w 0 , w 1 and nonzero real numbers a, b and c. As a special case, by taking the initial conditions 0, 1 and 2, b we define the sequences {u n } and {v n }, respectively. The main purpose of this study is to derive some basic properties of the sequences {u n }, {v n } and {w n } by using a matrix approach.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Some basic properties of the generalized bi-periodic Fibonacci and Lucas sequences

Tan

Leung

2020

Adv Differ Equ

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“…Gould's paper [4] serves as an excellent reference for the history of the method of Fibonacci Q-matrix. The matrix method is further explored by Bicknell and Hoggatt [5], Deveci [2], Khmovsky [7], Ekin and Tan [9], Tan [10], Waddill [12] to derive identities for the Fibonacci numbers and its generalizations. Bacon, Cook, and Graves [1] gave a recent account of a generalization of this method.…”

Section: Introductionmentioning

confidence: 99%

Some binomial-sum identities for the generalized bi-periodic Fibonacci sequences

Leung¹

2020

NNTDM

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A bi-periodic sequence is a sequence which satisfies different recurrence relations depending on whether the n-th term considered is odd or even. In this paper, we investigate the properties of the generalized bi-periodic Fibonacci sequences. It is a generalization of the biperiodic Fibonacci sequences defined by Edson and Yayenie. We derive binomial-sum identities for the generalized bi-periodic Fibonacci sequences by matrix method. Our identities generalize binomial-sum identities derived by Edson and Yayenie for the case of bi-periodic Fibonacci sequences.

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“…It is worthwhile to note that the research on generating new identities and summation formulas for various generalizations of the classical Fibonacci sequence by using matrix methods has been extensive recently. For examples, the work done by Ekin and Tan [4], Keskin and Siar [7], and Tan [9] serve as good references to the subject.…”

Section: Introductionmentioning

confidence: 99%