Generalized Lucas Numbers and Relations with Generalized Fibonacci Numbers

In this paper, we present a new generalization of the Lucas numbers by matrix representation using Genaralized Lucas Polynomials. We give some properties of this new generalization and some relations between the generalized order-k Lucas numbers and generalized order-k Fibonacci numbers. In addition, we obtain Binet formula and combinatorial representation for generalized order-k Lucas numbers by using properties of generalized Fibonacci numbers

Determinant and Permanent of Hessenberg Matrix and Generalized Lucas Polynomials

In this paper, we give some determinantal and permanental representations of Generalized Lucas Polynomials by using various Hessenberg matrices, which are general form of determinantal and permanental representations of ordinary Lucas and Perrin sequences. Then we show, under what conditions that the determinants of the Hessenberg matrix becomes its permanents

Simson Identity of Generalized m-step Fibonacci Numbers

One of the best known and oldest identities for the Fibonacci sequence $F_n$ is $F_{n+1}F_{n-1}-F_{n}^2=(-1)^n$ which was derived first by R. Simson in 1753 and it is now called as Simson or Cassini Identity. In this paper, we generalize this result to generalized m-step Fibonacci numbers and give an attractive formula. Furthermore, we present some Simson's identities of particular generalized m-step Fibonacci sequences

k Sequences of Generalized Van der Laan and Generalized Perrin Polynomials

In this paper, we present k sequences of Generalized Van der Laan Polynomials and Generalized Perrin Polynomials using Genaralized Fibonacci and Lucas Polynomials. We give some properties of these polynomials. We also obtain generalized order-k Van der Laan Numbers, k sequences of generalized order-k Van der Laan Numbers, generalized order-k Perrin Numbers and k sequences of generalized order-k Perrin Numbers. In addition, we examine the relationship between them

On the g-Circulant Matrix involving the Generalized k-Horadam Numbers

In this study, we present a new generalization of circulant matrices for the generalized $k$-Horadam numbers, by considering the $g$-circulant matrix $C_{n,g}(H)=g -circ(H_{k,1},H_{k,2},\ldots ,H_{k,n})$. Also, we calculate the spectral norm, determinant and inverse of $C_{n,g}(H)$ in such matrices having the elements of all second order sequences

Gaussian Generalized Tetranacci Numbers

In this paper, we define Gaussian generalized Tetranacci numbers and as special cases, we investigate Gaussian Tetranacci and Gaussian Tetranacci-Lucas numbers with their properties

Tribonacci and Tribonacci-Lucas Matrix Sequences with Negative Subscripts

In this paper, we define Tribonacci and Tribonacci-Lucas matrix sequences with negative indices and investigate their properties.Comment: arXiv admin note: substantial text overlap with arXiv:1809.0780

Applications of some special numbers obtained from a difference equation of degree three

In this paper we present applications of some special numbers obtained from a difference equation of degree three, especially in the Coding Theory. As a particular case, we obtain the generalized Pell-Fibonacci-Lucas numbers, which were extended to the generalized quaternions

Third-order Jacobsthal Generalized Quaternions

In this paper, the third-order Jacobsthal generalized quaternions are introduced. We use the well-known identities related to the third-order Jacobsthal and third-order Jacobsthal-Lucas numbers to obtain the relations regarding these quaternions. Furthermore, the third-order Jacobsthal generalized quaternions are classified by considering the special cases of quaternionic units. We derive the relations between third-order Jacobsthal and third-order Jacobsthal-Lucas generalized quaternions.Comment: Submitted to Journal. 16 page

Some remarks regarding a, b, x0, x1 numbers and a, b, x0, x1 quaternions

In this paper we define and study properties and applications of a, b, x0, x1 elements in some special cases