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In this paper, we solve in closed-form the following third-order system of nonlinear difference equations xn+1 = ynyn−1x p n−1/xn(any q n−2 +bnynyn−1),/ yn+1 = xnxn−1y q n−1 yn(cnx p n−2 +dnxnxn−1), p,q ∈ N,n ∈ N0 where the initial values x−i, y−i,i = 0,1,2 and the parameters (an)n∈N0,(bn)n∈N0,(cn)n∈N0, (dn)n∈N0 are non-zero real numbers. The form of the solutions of the one dimensional case of our system and a more general system defined by one to one functions are also presented. [ABSTRACT FROM AUTHOR]

In this paper, new types of k-Fibonacci numbers are described, with respect to the definition of the distance between numbers using a recurrence relation. However, these sequences differ not only by the value of the natural number k but also according to the value of a new parameter r which is used in defining this distance. Furthermore, various properties of these new numbers are discussed. In the second part of this paper, we apply the binomial transform to these generalized k-Fibonacci sequences. [ABSTRACT FROM AUTHOR]

In this paper, we define Gaussian Fibonacci quaternion polynomials and Gaussian Lucas quaternion polynomials. We also investigate some properties of these quaternion polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Diophantine equation Ln±Lm = kx² with k ∈ {1,2} and we find all solutions of this equation in nonnegative integers n,m, and x when n ≡ m(mod2). With the help of these solutions, we solve the equation Ln -Lm = 2a. In order to solve the last equation, we also use lower bounds for linear forms in logarithms and a version of the Baker-Davenport reduction method in diophantine approximation. [ABSTRACT FROM AUTHOR]

This paper deals with developing a new class of quaternions, called hyper-dual Horadam quaternions which are constructed from the quaternions whose components are hyper-dual Horadam numbers. We investigate the algebraic properties of these quaternions. [ABSTRACT FROM AUTHOR]

For an integer k ≥ 2, let (Fn(k))n be the k-Fibonacci sequence which starts with 0,...0,1 (k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we search for powers of 2 which are sums of two k-Fibonacci numbers. The main tools used in this work are lower bounds for linear forms in logarithms and a version of the Baker-Davenport reduction method in diophantine approximation. This paper continues and extends the previous work of [4] and [2]. [ABSTRACT FROM AUTHOR]

In this paper we represent the well-defined solutions of the system of the higher-order rational difference equations ... in terms of Fibonacci and Lucas sequences, where the initial values... j = 1, 2, ..., 2p + 1, do not equal -3. Some theoretical explanations related to the representation for the general solution are also given. [ABSTRACT FROM AUTHOR]

Hybrid numbers generalize complex, hyperbolic and dual numbers, simultaneously. Special kinds of hybrid numbers, related to numbers of Fibonacci type, among others Pell numbers, were introduced quite recently. In this paper we introduce and study polynomials, which are a generalization of Pell hybrid numbers and so called Pell hybrinomials. [ABSTRACT FROM AUTHOR]

In this paper we introduce the generalized bi-periodic Fibonacci and Lucas quaternions which are the further generalizations of the bi-periodic Fibonacci and Lucas quaternions considered in the literature. For those quaternions, we derive the generating functions, Binet's formulas and Catalan's identities. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce new families of the q-Fibonacci and q-Lucas polynomials, which are represented here as the incomplete q-Fibonacci polynomials Fnk (x, s, q) and the incomplete q-Lucas polynomials Lnk (x, s, q), respectively. These polynomials provide the q-analogues of the incomplete Fibonacci and Lucas numbers. We give several properties and generating functions of each of these families q-polynomials. We also point out the fact that the results for the q-analogues which we consider in this article for 0 < q < 1 can easily be translated into the corresponding results for the (p, q)-analogues (with 0 < q < p ≥ 1) by applying some obvious parametric variations, the additional parameter p being redundant. [ABSTRACT FROM AUTHOR]

In this paper, we determine the sums ... and some congruences can be obtained by using them. For example, for an odd prime p ≠ 5, ... and for an odd prime p, ... where d ∈ {0, ..., (p-1)/2}, Fn is the nth Fibonacci number, Ln is the nth Lucas number and Pn is the nth Pell number. (./p) denotes the Legendre symbol. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Fibonacci conditional sequence {fn} and the Lucas conditional sequence flng. We derive some properties of Fibonacci and Lucas conditional sequences by using the matrix method. Our results are elegant as the results for ordinary Fibonacci and Lucas sequences. [ABSTRACT FROM AUTHOR]

Let k≥3 be an odd integer. In this paper, we investigate all positive integer solutions of the equations x4-kx2y+y2=∓A, x4-kx2y+y2=∓A(k2-4), x4-(k2-4)y2=∓4A, and x2-(k2-4)y4=∓4A with A=∓(k∓2). We show that if k≡1(mod8) and k2-4 be a squarefree integer, then the equation x4-kx2y+y2=(k-2)(k2-4) has no positive integer solutions. Moreover, if k2-4 be a squarefree integer, then the equation x4-kx2y+y2=-(k+2)(k2-4) has no positive integer solutions. [ABSTRACT FROM AUTHOR]

Recently Melham [1] gives closed formulæ for certain finite reciprocal sums. In this paper, we present a different approach to compute these sums in closed form. Our approach is straight-forward and simple. First we convert the sums in q-notation, then use partial fraction decomposition and telescoping to derive closed formulæ. [ABSTRACT FROM AUTHOR]

The generalized bi-periodic Fibonacci sequence f{qn}∞n=0 have initial conditions q0=0, q1 = 1 and recurrence relation qn = qn-1 + qn-2 (when n is even and n ≥ 2) or qn = bqn-1 + qn-2 (when n is odd and n ≥ 3), where a and b are nonzero real numbers. Some well-known sequences, such as the Fibonacci sequence, the Pell sequence and the k-Fibonacci sequence, are special cases of this generalization. In this paper, we consider the partial infinite sum derived from the reciprocal of the generalized bi-periodic Fibonacci numbers. [ABSTRACT FROM AUTHOR]

In this paper, we consider Marques and Trojovsky's conjecture involving Fibonomial coefficients, Fibonacci and Lucas numbers. After rewriting it in terms of Gaussian q-binomial coefficients and q-Pochhammer symbols, we give a solution of the conjecture. The approach is to use q-analysis to prove the claimed formulas. [ABSTRACT FROM AUTHOR]

In this paper, we determine when the equation in the title has an infinite number of positive integer solutions x and y when 0 ≤ n ≤ 10. Moreover, we give all the positive integer solutions of the same equation for 0 ≤ n ≤ 10. [ABSTRACT FROM AUTHOR]

In this paper, we give general formulas for some weighted binomial sums, using the powers of terms of certain binary recurrences. As an applications of our results, we show that the weighted binomial sums of the generalized Fibonacci and Lucas numbers can be expressed via the second kinds of Chebyshev polynomials and Stirling numbers. [ABSTRACT FROM AUTHOR]

In this paper we study the relationship between the generalized Lucas polynomials and the Euler numbers and give several interesting identities involving them. [ABSTRACT FROM AUTHOR]