Kemal Uslu

A new Fibonacci type collocation procedure for boundary value problems

In this study, we present a new procedure for the numerical solution of boundary value problems. This approach is mainly founded on the Fibonacci polynomial expansions, the so-called pseudospectral methods with the collocation method. The applicability and effectiveness of our proposed approach is shown by some illustrative examples. Then, the results indicate that this method is very effective and highly promising for linear differential equations defined on any subinterval of the real domain.MSC: 35A25.

A new approach to generalized Fibonacci and Lucas numbers with binomial coefficients

In this study, Fibonacci and Lucas numbers have been obtained by using generalized Fibonacci numbers. In addition, some new properties of generalized Fibonacci numbers with binomial coefficients have been investigated to write generalized Fibonacci sequences in a new direct way. Furthermore, it has been given a new formula for some Lucas numbers.

On the properties of Lucas numbers with binomial coefficients

In this study, some new properties of Lucas numbers with binomial coefficients have been obtained to write Lucas sequences in a new direct way. In addition, some important consequences of these results related to the Fibonacci numbers have been given.

The Generalized (s,t)‐Sequence and its Matrix Sequence

In this study, we first define a new sequence in which it generalizes (s,t)‐Fibonacci and (s,t)‐Lucas sequences at the same time. After that, by using it, we establish generalized (s,t)‐matrix sequences. Finally we present some important relationships among this new generalization, (s,t)‐Fibonacci and (s,t)‐Lucas sequences and their matrix sequences.

The periodicity and solutions of the rational difference equation with periodic coefficients

Abstract In this paper, we give necessary and sufficient conditions for generalized solution and periodicity of the difference equation x n + 1 = p n x n − k + x n − ( k + 1 ) q n + x n − ( k + 1 ) with ( k + 2 ) -periodic coefficients, where k ∈ N , x − k − 1 , x − k , ⋯ , x 0 ∈ R . Also, we obtain that the generalized solution is periodic with ( k + 1 ) -period.

On the Binomial Sums of k‐Fibonacci and k ‐Lucas sequences

The main purpose of this paper is to establish some new properties of k‐Fibonacci and k‐Lucas numbers in terms of binomial sums. By that, we can obtain these special numbers in a new and direct way. Moreover, some connections between k‐Fibonacci and k‐Lucas numbers are revealed to get a more strong result.

The Construction of Horadam Numbers in Terms of the Determinant of Tridiagonal Matrices

In this study, by using determinants of tridiagonal matrices, we mainly obtain Horadam numbers with positive and negative indices. Therefore we establish a new generalization for the tridiagonal matrices that represent well known numbers such as Fibonacci, Lucas, Jacobsthal, Jacobsthal‐Lucas, Pell and Pell‐Lucas numbers.

The (s, t)-Generalized Jacobsthal Matrix Sequences

In this study, we consider sequences named (s, t)-Jacobsthal, (s, t)-Jacobsthal–Lucas and defined generalized (s, t)-Jacobsthal integer sequences. After that, by using these sequences, we define generalized (s, t)-Jacobsthal matrix sequence in which it generalizes (s, t)-Jacobsthal matrix sequence, (s, t)-Jacobsthal–Lucas matrix sequence at the same time. Finally we investigate some properties of the sequence and present some important relationship among (s, t)-Jacobsthal matrix sequence, (s, t)-Jacobsthal–Lucas matrix sequence and generalized (s, t)-Jacobsthal matrix sequence.

Erratum to “On the norms of circulant matrices with the Fibonacci and Lucas numbers” [Appl. Math. Comput. 160 (1) (2005) 125–132]

A compressed-gas circuit breaker possesses two contact members (1, 2), which move relative to one another, and a nozzle (7), which is made of dielectric material and is attached to a first (2) member of the two contact members. The constriction (8) of the nozzle (7) which is made of dielectric material, separates a compression-space (9) from an expansion-space (15), and the compressed gas used for extinguishing the arc which occurs on operating the circuit breaker flows through this constriction. An insert (13), designed in the shape of an annulus, is provided at the nozzle constriction (8). The design of the nozzle (7), which is made of dielectric material, is such that it can be exposed to a high thermal loading in the region of its constriction, while, at the same time, high voltages can be held in the breaker-gap, without the occurrence of instances of arcing-over. This object is achieved by arranging the insert (13) to be electrically isolated with respect to the two contact members (1, 2) the insert exhibits a first capacitance with respect to the first contact member (2) and a second capacitance with respect to the second contact member (1). The magnitudes of the capacitances are chosen, by suitable arrangement and dimensioning of the insert (13), so that the electric field in the region of the nozzle constriction (8) is displaced at least partially from the surface of the nozzle, into the compression-space (9) and into the expansion-space (15).