On some class of periodic-discrete homogeneous difference equations via Fibonacci sequences

Taher, R. Ben; Benkhaldoun, H.; Rachidi, M.

doi:10.1080/10236198.2016.1194407

Cited by 4 publications

(5 citation statements)

References 9 publications

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“…Expressions of type (3) represent a linear difference equation of the second order with variable coefficients. Several methods and techniques have been developed to solve linear difference equations with variable coefficients (see, for instance, [1,7,12,[14][15][16]20], and references therein). This topic continues to attract much attention due to their many applications in mathematics and applied sciences (see for instance [1,2,6,7]).…”

Section: Introductionmentioning

confidence: 99%

“…Several methods and techniques have been developed to solve linear difference equations with variable coefficients (see, for instance, [1,7,12,[14][15][16]20], and references therein). This topic continues to attract much attention due to their many applications in mathematics and applied sciences (see for instance [1,2,6,7]). In [20], Popenda gave explicit formulas for the general solutions of homogeneous and non-homogeneous second-order linear difference equations, with arbitrarily varying coefficients, using a direct computation.…”

Section: Introductionmentioning

confidence: 99%

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New approaches of (q,k)-Fibonacci–Pell sequences via linear difference equations. Applications

Craveiro,

Pereira Spreafico,

Rachidi

2023

NNTDM

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In this paper we establish some explicit formulas of (q,k)-Fibonacci–Pell sequences via linear difference equations of order 2 with variable coefficients, and explore some of their new properties. More precisely, our results are based on two approaches, namely, the determinantal and the nested sums approaches, and their closed relations. As applications, we investigate the q-analogue Cassini identities and examine a pair of Rogers–Ramanujan type identities.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

New approaches of (q,k)-Fibonacci–Pell sequences via linear difference equations. Applications

Craveiro,

Pereira Spreafico,

Rachidi

2023

NNTDM

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“…(1.1) (see, for example, [15,17], and references therein). Recently, the homogeneous linear difference equations (1.1) with periodic coefficients, i.e., a j (n + p) = a j (n), have been solved in [4,5], using properties of the generalized Fibonacci sequences in the algebra of square matrices. More precisely, in [4], Eq.…”

Section: Introductionmentioning

confidence: 99%

“…In the last years, the product of companion matrices has attracted much attention, because this product occurs in various fields of mathematics and applied sciences, such that the Floquet system theory related to the linear difference equations (see [4,5,16,17]). Diverse methods for computing the product of companion matrices have been proposed in the literature.…”

Section: Introductionmentioning

confidence: 99%

“…Another expression for the product of companion matrices was obtained in [17], based on the study of solutions of non-homogeneous and homogeneous linear difference equations of order N, with variable coefficients. Recently, it was shown in [4,5] that the product of companion matrices plays a central role, for investigating a large class of periodic-discrete homogeneous difference equations via generalized Fibonacci sequences. Moreover, through the key of generalized Fibonacci sequences, there are still some interesting and relevant problems that can be examined.…”

Section: Introductionmentioning

confidence: 99%

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Periodic matrix difference equations and companion matrices in blocks: some applications

Benkhaldoun¹,

Taher²,

Rachidi³

2021

Arab. J. Math.

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This study is devoted to some periodic matrix difference equations, through their associated product of companion matrices in blocks. Linear recursive sequences in the algebra of square matrices in blocks and the generalized Cayley–Hamilton theorem are considered for working out some results about the powers of matrices in blocks. Two algorithms for computing the finite product of periodic companion matrices in blocks are built. Illustrative examples and applications are considered to demonstrate the effectiveness of our approach.

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New method for solving non-homogeneous periodic second-order difference equation and some applications

Taher

Lassri

Rachidi³

2023

Arab. J. Math.

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In the present study, we are interested in solving the nonhomogeneous second-order linear difference equation with periodic coefficients of period $$ p\ge 2$$ p ≥ 2 , by bringing two new approaches enabling us to provide both analytic and combinatorial solutions to this family of equations. First, we get around the problem by converting this kind of equations to an equivalent family of nonhomogeneous linear difference equations of order p with constant coefficients. Second, we propose new expressions of the solutions of this family of equations, using our techniques of calculating the powers of product of companion matrices and some properties of generalized Fibonacci sequences. The study of the special case $$ p=2 $$ p = 2 is provided. And to enhance the effectiveness of our approaches, some numerical examples are discussed.

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On some class of periodic-discrete homogeneous difference equations via Fibonacci sequences

Cited by 4 publications

References 9 publications

New approaches of (q,k)-Fibonacci–Pell sequences via linear difference equations. Applications

New approaches of (q,k)-Fibonacci–Pell sequences via linear difference equations. Applications

Periodic matrix difference equations and companion matrices in blocks: some applications

New method for solving non-homogeneous periodic second-order difference equation and some applications

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