Pauli–Fibonacci quaternions

The aim of this work is to consider the Pauli–Fibonacci quaternions and to present some properties involving this sequence, including the Binet’s formula and generating functions. Furthermore, the Honsberger identity, the generating function, d’Ocagne’s identity, Cassini’s identity, Catalan’s identity for these quaternions are given. The matrix representations for Pauli–Fibonacci quaternions are introduced.

Symmetric and generating functions of generalized (p,q)-numbers

In this paper, we will firstly define a new generalization of numbers (p, q) and then derive the appropriate Binet's formula and generating functions concerning (p,q)-Fibonacci numbers, (p,q)- Lucas numbers, (p,q)-Pell numbers, (p,q)-Pell Lucas numbers, (p,q)-Jacobsthal numbers and (p,q)-Jacobsthal Lucas numbers. Also, some useful generating functions are provided for the products of (p,q)-numbers with bivariate complex Fibonacci and Lucas polynomials.

On the k-Mersenne–Lucas numbers

In this paper, we will introduce a new definition of k-Mersenne–Lucas numbers and investigate some properties. Then, we obtain some identities and established connection formulas between k-Mersenne–Lucas numbers and k-Mersenne numbers through the use of Binet’s formula.

On J(r, n)-Jacobsthal Hybrid Numbers

In this paper we introduce a generalization of Jacobsthal hybrid numbers – J(r, n)-Jacobsthal hybrid numbers. We give some of their properties: character, Binet’s formula, a summation formula and a generating function.

Dual bicomplex Horadam quaternions

The aim of this work is to introduce a generalization of dual quaternions called dual bicomplex Horadam quaternions and to present some properties, the Binet’s formula, Catalan’s identity, Cassini’s identity and the summation formula for this type of bicomplex quaternions. Furthermore, several identities for dual bicomplex Fibonacci quaternions are given.

On some properties of split Horadam quaternions

In this paper we introduce and study the split Horadam quaternions. We give some identities, among others Binet’s formula, Catalan’s, Cassini’s and d’Ocagne’s identities for these numbers.

The Hybrid Numbers of Padovan and Some Identities

In this article, we will define Padovan’s hybrid numbers, based on the new noncommutative numbering system studied by Özdemir ([7]). Such a system that is a set involving complex, hyperbolic and dual numbers. In addition, Padovan’s hybrid numbers are created by combining this set, satisfying the relation ih = −hi = ɛ + i. Given this, some properties and identities are shown for these numbers, such as Binet’s formula, generating matrix, characteristic equation, norm, and generating function. In addition, these numbers are extended to the integer field and some identities are made.

On a New One Parameter Generalization of Pell Numbers

In this paper we present a new one parameter generalization of the classical Pell numbers. We investigate the generalized Binet’s formula, the generating function and some identities for r-Pell numbers. Moreover, we give a graph interpretation of these numbers.

Sebuah Generalisasi Baru Barisan Fibonacci-Lucas

Barisan Fibonacci dan Lucas telah digeneralisasi dalam banyak cara, beberapa dengan mempertahankan kondisi awal, dan lainnya dengan mempertahankan relasi rekurensi. Makalah ini menyajikan sebuah generalisasi baru barisan Fibonacci-Lucas yang didefinisikan oleh relasi rekurensi B_n=B_(n-1)+B_(n-2),n≥2 , B_0=2b,B_1=s dengan b dan s bilangan bulat tak negatif. Selanjutnya, beberapa identitas dihasilkan dan diturunkan menggunakan formula Binet dan metode sederhana lainnya. Juga dibahas beberapa identitas dalam bentuk determinan. The Fibonacci and Lucas sequence has been generalized in many ways, some by preserving the initial conditions, and others by preserving the recurrence relation. In this paper, a new generalization of Fibonacci-Lucas sequence is introduced and defined by the recurrence relation B_n=B_(n-1)+B_(n-2),n≥2, with , B_0=2b,B_1=s where b and s are non negative integers. Further, some identities are generated and derived by Binet’s formula and other simple methods. Also some determinant identities are discussed.

On the Bicomplex Generalized Tribonacci Quaternions

In this paper, we introduce the bicomplex generalized tribonacci quaternions. Furthermore, Binet’s formula, generating functions, and the summation formula for this type of quaternion are given. Lastly, as an application, we present the determinant of a special matrix, and we show that the determinant is equal to the n th term of the bicomplex generalized tribonacci quaternions.