Necati Taskara

On the solutions of two special types of Riccati difference equation via Fibonacci numbers

In this study, we investigate the solutions of two special types of the Riccati difference equation xn+1=11+xn and yn+1=1−1+yn such that their solutions are associated with Fibonacci numbers.MSC: 11B39, 39A10, 39A13.

On fourteen solvable systems of difference equations

Abstract In this paper, we mainly consider the systems of difference equations x n + 1 = 1 + p n q n , y n + 1 = 1 + r n s n , n ∈ N 0 , where each of the sequences p n , q n , r n and s n represents either the sequence x n or the sequence y n , with nonzero real initial values x 0 and y 0 . Then we solve fourteen out of sixteen possible systems. It is noteworthy to depict that the solutions are presented in terms of Fibonacci numbers for twelve systems of these fourteen systems.

On the (s,t)-Pell and (s,t)-Pell–Lucas sequences and their matrix representations

Abstract In this paper, we first give new generalizations for ( s , t ) -Pell { p n ( s , t ) } n ∈ N and ( s , t ) -Pell Lucas { q n ( s , t ) } n ∈ N sequences for Pell and Pell–Lucas numbers. Considering these sequences, we define the matrix sequences which have elements of { p n ( s , t ) } n ∈ N and { q n ( s , t ) } n ∈ N . Then we investigate their properties.

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 norms of an r-circulant matrix with the generalized k-Horadam numbers

In this paper, we present new upper and lower bounds for the spectral norm of an r-circulant matrix H=Cr(Hk,0,Hk,1,Hk,2,…,Hk,n−1) whose entries are the generalized k-Horadam numbers. Furthermore, we obtain new formulas to calculate the eigenvalues and determinant of the matrix H.MSC:11B39, 15A60, 15A15.

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.

Matrix Sequences in terms of Padovan and Perrin Numbers

The first main idea of this paper is to develop the matrix sequences that represent Padovan and Perrin numbers. Then, by taking into account matrix properties of these new matrix sequences, some behaviours of Padovan and Perrin numbers will be investigated. Moreover, some important relationships between Padovan and Perrin matrix sequences will be presented.

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.

Singular value inequalities for the arithmetic, geometric and Heinz means of matrices

Let A and B be positive definite n × n matrices such that A ≥ B. Among other results, it is shown that and for j = 1, 2, … , n, where A♯1/2 B is the geometric mean of A and B and s j (X), j = 1, 2, … , n, are the singular values of an n × n matrix X.

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.