On the matrix powers and exponential by the r-generalized Fibonacci sequences methods: the companion matrix case

Taher, R. Ben; Rachidi, M.

doi:10.1016/s0024-3795(03)00418-x

Cited by 30 publications

(11 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It seems that (2.7) is not as widely known as it should be. In [1,2] an equivalent formula, originally obtained from properties of linearly recurrent sequences, is used to find expressions for the powers and the exponential of a matrix. In [24] the formula is also used to find an expression for A m equivalent to our formula (3.4).…”

Section: Final Remarksmentioning

confidence: 99%

Functions of matrices

Verde‐Star

2005

Linear Algebra and its Applications

View full text Add to dashboard Cite

We use the Cayley-Hamilton theorem and the sequence of Horner polynomials associated with a polynomial w(z) to obtain explicit formulas for functions of the form f (tA), where f is defined by a convergent power series and A is a square matrix. We use a well-known explicit formula for the resolvent of A and show that f (tA) is the Hadamard product of f (t) and the function (I − tA) −1 , which is easily obtained from the resolvent.

show abstract

Section: Final Remarksmentioning

confidence: 99%

Functions of matrices

Verde‐Star

2005

Linear Algebra and its Applications

View full text Add to dashboard Cite

show abstract

“…It was established in that the Chen-Louck Theorems [5, Theorem 3.1] can be recovered by a direct computation from Expression (22) of [2]. In [8] the authors recall the Chen-Louck Theorem (see Theorem 5 of [8]), and formulate the combinatorial expression of the generalized Pell numbers P We conclude this section by observing that Expression (4.1) can be written under the form…”

Section: Combinatorial Identities For the Pell Fundamental System P Rmentioning

confidence: 98%

“…(n) i j of the powers A n are expressed in terms of a family of generalized Fibonacci sequences v [2], page 348). Thus, we can show that Formula (4) of [8] represents a particular case of the preceding studies.…”

Section: Pell Fundamental System P R and Some Generalized Pell Identitiesmentioning

confidence: 99%

On Generalized Pell Numbers of Order r ≥ 2

Spreafico

Rachidi

2021

TEMA

View full text Add to dashboard Cite

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.

show abstract

“…Fibonacci numbers are one of the most well-known numbers, and it has many important applications to diverse fields such as mathematics, computer science, physics, biology and statistics. We can see applications of Fibonacci sequences in group theory in (C. Campbell and P. Campbell, 2005), (Deveci, 2011(Deveci, , 2015 and also see some generalized Fibonacci and Lucas sequences in (Arı 2015), (Grabowski and Wojtecki, 2004), (İpek and Arı, 2015), (Kılıç and Taşçı, 2006), (Lee, 2000), (P. Stanimirovic, Nikolov and I. Stanimirovic 2008), (Taher and Rachidi, 2003), (Taşçı and Kılıç, 2004). Y. Taşyurdu and İ. Gültekin obtain the period of generalized Fibonacci sequence in finite rings with identity of order Erzincan Univ J Sci Tech 9(1), 2016, 95-101 Taşyurdu ve ark.…”

Section: Introductionmentioning

confidence: 97%

“…Many of the obtained numbers by using homogeneous lineer recurrence relations and their the miscellaneous properties have been stuied; see, for example, (Lee, 2000), (Taher and Rachidi, 2003). Fibonacci numbers are one of the most well-known numbers, and it has many important applications to diverse fields such as mathematics, computer science, physics, biology and statistics.…”

Section: Introductionmentioning

confidence: 99%