Yasin Yazlik

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 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.

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.

On the spectral norms of r-circulant matrices with the biperiodic Fibonacci and Lucas numbers

In this paper, we present new upper and lower bounds for the spectral norms of the r-circulant matrices Q=Cr((ba)ξ(1)2q0,(ba)ξ(2)2q1,(ba)ξ(3)2q2,…,(ba)ξ(n)2qn−1)

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

Q=C_{r} ( (\frac{b}{a} )^{\frac{\xi (1)}{2}}q_{0}, (\frac{b}{a} )^{\frac{\xi(2)}{2}}q_{1}, (\frac {b}{a} )^{\frac{\xi(3)}{2}}q_{2}, \dots, (\frac{b}{a} )^{\frac{\xi(n)}{2}}q_{n-1} )

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

and L=Cr((ba)ξ(0)2l0,(ba)ξ(1)2l1,(ba)ξ(2)2l2,…,(ba)ξ(n−1)2ln−1)

A note on generalized k-Horadam sequence

L=C_{r} ( (\frac {b}{a} )^{\frac{\xi(0)}{2}}l_{0}, (\frac{b}{a} )^{\frac{\xi (1)}{2}}l_{1}, (\frac{b}{a} )^{\frac{\xi(2)}{2}}l_{2}, \dots, (\frac{b}{a} )^{\frac{\xi(n-1)}{2}}l_{n-1} )

On the solutions of a max-type difference equation system

whose entries are the biperiodic Fibonacci and biperiodic Lucas numbers, respectively. Finally, we obtain lower and upper bounds for the spectral norms of Kronecker and Hadamard products of Q and L matrices.

Spectral norm, Eigenvalues and Determinant of Circulant Matrix involving the Generalized k-Horadam numbers.

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.