Generalized Fibonacci Numbers Research Articles

New Properties and Matrix Representations on Higher-Order Generalized Fibonacci Quaternions with q-Integer Components

In this paper, by using q-integers and higher-order generalized Fibonacci numbers, we define the higher-order generalized Fibonacci quaternions with q-integer components. We give some special cases of these newly established quaternions. This article examines q-calculus and quaternions together. We obtain a Binet-like formula, some new identities, a generating function, a recurrence relation, an exponential generating function, and sum properties of quaternions with quantum integer coefficients. In addition, we obtain some new identities for these types of quaternions by using three new special matrices.

Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices

By considering the tiling of an N-board (a linear array of N square cells of unit width) with new types of tile that we refer to as combs, we give a combinatorial interpretation of the product of two consecutive generalized Fibonacci numbers s n (where s n = ∑ i = 1 q v i s n − m i , s 0 = 1 , s n < 0 = 0 , where v i and m i are positive integers and m 1 < ⋯ < m q ) each raised to an arbitrary non-negative integer power. A ( w , g ; m ) -comb is a tile composed of m rectangular sub-tiles of dimensions w × 1 separated by gaps of width g. The interpretation is used to give combinatorial proof of new convolution-type identities relating s n 2 for the cases q = 2, v i = 1 , m 1 = M , m 2 = m + 1 for M = 0, m to the permanent of a (0,1) Toeplitz matrix with 3 nonzero diagonals which are − 2 , M−1, and m above the leading diagonal. When m = 1, these identities reduce to ones connecting the Padovan and Narayana's cows numbers.

On 3-parameter quaternions with higher order generalized Fibonacci numbers components

On 3-parameter quaternions with higher order generalized Fibonacci numbers components

Dual quaternion theory over HGC numbers

Knowing the applications of quaternions in various fields, such as robotics, navigation, computer visualization and animation, in this study, we give the theory of dual quaternions considering Hyperbolic-Generalized Complex (HGC) numbers as coefficients via generalized complex and hyperbolic numbers. We account for how HGC number theory can extend dual quaternions to HGC dual quaternions. Some related theoretical results with HGC Fibonacci/Lucas numbers are established, including their dual quaternions. Given HGC Fibonacci/Lucas numbers, their special matrix correspondences have been identified and these are carried out to HGC Fibonacci/Lucas dual quaternions. Furthermore, we provide a more accurate way to quickly calculate HGC Fibonacci numbers and associate this with HGC generalized Fibonacci numbers. For implementation, we produce an algorithm in Maple. Lastly, we put the theory into practice.

Arithmetic properties of generalized Fibonacci numbers

Arithmetic properties of generalized Fibonacci numbers

On a Solvable System of Difference Equations in Terms of Generalized Fibonacci Numbers

ABSTRACT In this paper, we represent that the following three-dimensional system of difference equations x n + 1 = α y n + a y n y n − β z n − 1 , y n + 1 = β z n + b z n z n − γ x n − 1 , z n + 1 = γ x n + c x n x n − α y n − 1 , n ∈ ℕ 0 , $$\matrix{{{x_{n + 1}} = \alpha {y_n} + {{a{y_n}} \over {{y_n} - \beta {z_{n - 1}}}},\quad {y_{n + 1}} = \beta {z_n} + {{b{z_n}} \over {{z_n} - \gamma {x_{n - 1}}}},\quad {z_{n + 1}} = \gamma {x_n} + {{c{x_n}} \over {{x_n} - \alpha {y_{n - 1}}}},\qquad n \in {{\mathbb N}_0},} \cr} $$ where the parameters a, b, c, α, β, γ and the initial values x −i , y −i , z −i , i ∈ {0, 1}, are real numbers, can be solved in closed form by using transformation. We analyzed the solutions in 10 different cases depending on whether the parameters are zero or nonzero. It is noteworthy to depict that the solutions of some particular cases of this system are presented in terms of generalized Fibonacci numbers. Note that our results considerably extend and improve some recent results in the literature.

On $k$-generalized Lucas sequence with its triangle

In this paper, we investigate several identities of $k$-generalized Lucas numbers with $k$-generalized Fibonacci numbers. We also establish a link between generalized $s$-Lucas triangle and bi$^{s}$nomial coefficients given by the coefficients of the development of a power of $(1+x+x^{2}+\cdots+x^{s}),$ with $s \in \mathbb{N}.$

GCD of Sums of k Consecutive Squares of Generalized Fibonacci Numbers

GCD of Sums of k Consecutive Squares of Generalized Fibonacci Numbers

Ratios of Generalized Fibonacci Numbers

Ratios of Generalized Fibonacci Numbers

Gray codes for Fibonacci q-decreasing words

A length n binary word is q-decreasing, q⩾1, if every of its maximal factors of the form 0a1b satisfies a=0 or q⋅a>b. In particular, in 1-decreasing words every run of 0s is immediately followed by a strictly shorter run of 1s. We show constructively that these words are in bijection with binary words having no occurrences of 1q+1, and thus they are enumerated by the (q+1)-generalized Fibonacci numbers. We give some enumerative results and reveal similarities between q-decreasing words and binary words having no occurrences of 1q+1 in terms of the frequency of 1-bits. In the second part of our paper, we provide an efficient exhaustive generating algorithm for q-decreasing words in lexicographic order, for any q⩾1, show the existence of 3-Gray codes and explain how a generating algorithm for these Gray codes can be obtained. Moreover, we give the construction of a more restrictive 1-Gray code for 1-decreasing words, which in particular settles a conjecture stated recently in the context of interconnection networks by Eğecioğlu and Iršič.

Some equalities and binomial sums about the generalized Fibonacci number $u_n$

In this study, we take the generalized Fibonacci sequence \{u_{n}\} as u_{0}=0,u_{1}=1 and \ u_{n}=ru_{n-1}+u_{n-2} for n&gt;1, where r is a non-zero integer. Based on Halton’s paper in [4], we derive three interrelated functions involving the terms of generalized Fibonacci sequence \{u_{n}\}. Using these three functions we introduce a simple approach to obtain a lot of identities, binomial sums and alternate binomial sums involving the terms of generalized Fibonacci sequence \{u_{n}\}.

“Generating matrix for Generalized Fibonacci numbers and Fibonacci polynomials

Many researchers have been working on recurrence relation sequences of numbers and polynomials which are useful topic not only in mathematics but also in physics, economics and various applications in many other fields. There are many useful identities on recurrence relation sequence but there main problem to find any term of recurrence relation sequence we need to find previous all terms of recurrence relation sequence of numbers and polynomials. There were many important theorems obtained on recurrence relation sequences. In this paper we have given special identity for generalized Fibonacci sequence of number and Fibonacci sequence of polynomials. These identities are very useful to represent Fibonacci generalized sequence of numbers and Fibonacci sequence of polynomials in the form of matrix. Authors define a special formula in this paper by this we can find special representation of Fibonacci generalized sequence of numbers and Fibonacci sequence of polynomials in the form of matrix. So, we can say that this paper is generalization of property of Fibonacci sequence of number and Fibonacci sequence of polynomials

On the self-convolution of generalized Fibonacci numbers

We focus on a family of equalities pioneered by Zhang and generalized by Zao and Wang and hence by Mansour which involves self convolution of generalized Fibonacci numbers. We show that all these formulas are nicely stated in only one equation involving a bivariate ordinary generating function and we give also a formula for the coefficients appearing in that context. As a consequence, we give the general forms for the equalities of Zhang, Zao-Wang and Mansour.

On the sum of the reciprocals of k-generalized Fibonacci numbers

Abstract In this note, we that if { F n ( k ) } n ≥ 0 {\left\{ {F_n^{\left( k \right)}} \right\}_{n \ge 0}} denotes the k-generalized Fibonacci sequence then for n ≥ 2 the closest integer to the reciprocal of ∑ m ≥ n 1 / F m ( k ) \sum\nolimits_{m \ge n} {1/F_m^{\left( k \right)}} is F n ( k ) − F n − 1 ( k ) F_n^{\left( k \right)} - F_{n - 1}^{\left( k \right)} .

$k$-Fibonacci numbers and $k$-Lucas numbers and associated bipartite graphs

In [6], [8] and [10], the authors studied the generalized Fibonacci numbers. Also, in [7], the author found a class of bipartite graphs whose number of $1$-factors is the $n$th $k$-Lucas numbers. In this paper, we give a new relationship between $g_n^{(k)}$ and $l_n^{(k)}$ and the number of $1$-factors of a bipartite graph.

MacMahon Partition Analysis: A discrete approach to broken stick problems

We propose a discrete approach to solve problems on forming polygons from broken sticks, which is akin to counting polygons with sides of integer length subject to certain Diophantine inequalities. Namely, we use MacMahon's Partition Analysis to obtain a generating function for the size of the set of segments of a broken stick subject to these inequalities. In particular, we use this approach to show that for n≥k≥3, the probability that a k-gon cannot be formed from a stick broken into n parts is given by n! over a product of linear combinations of partial sums of generalized Fibonacci numbers, a problem which proved to be very hard to generalize in the past.

Curious Generalized Fibonacci Numbers

A generalization of the well-known Fibonacci sequence is the k−Fibonacci sequence whose first k terms are 0,…,0,1 and each term afterwards is the sum of the preceding k terms. In this paper, we find all k-Fibonacci numbers that are curious numbers (i.e., numbers whose base ten representation have the form a⋯ab⋯ba⋯a). This work continues and extends the prior result of Trojovský, who found all Fibonacci numbers with a prescribed block of digits, and the result of Alahmadi et al., who searched for k-Fibonacci numbers, which are concatenation of two repdigits.

On some 3 × 3 dimensional matrices associated with generalized Fibonacci numbers

In this work, it is presented a procedure to find some 3 × 3 dimensional matrices whose integer powers can be characterized by generalized Fibonacci numbers. Moreover, some numerical examples are given to exemplify the procedure established.

Quantum coin flipping, qubit measurement, and generalized Fibonacci numbers

The problem of Hadamard quantum coin measurement in $n$ trials, with arbitrary number of repeated consecutive last states is formulated in terms of Fibonacci sequences for duplicated states, Tribonacci numbers for triplicated states and $N$-Bonacci numbers for arbitrary $N$-plicated states. The probability formulas for arbitrary position of repeated states are derived in terms of Lucas and Fibonacci numbers. For generic qubit coin, the formulas are expressed by Fibonacci and more general, $N$-Bonacci polynomials in qubit probabilities. The generating function for probabilities, the Golden Ratio limit of these probabilities and Shannon entropy for corresponding states are determined. By generalized Born rule and universality of $n$-qubit measurement gate, we formulate problem in terms of generic $n$-qubit states and construct projection operators in Hilbert space, constrained on the Fibonacci tree of the states. The results are generalized to qutrit and qudit coins, described by generalized Fibonacci-$N$-Bonacci sequences.

NON-SPECTRAL PROBLEM FOR CANTOR MEASURES

The spectral and non-spectral problems of measures have been considered in recent years. For the Cantor measure [Formula: see text], Hu and Lau [Spectral property of the [Formula: see text]ernoulli convolutions, Adv. Math. 219(2) (2008) 554–567] showed that [Formula: see text] contains infinite orthogonal exponentials if and only if [Formula: see text] becomes some type of binomial number. In this paper, we classify the spectral number of the Cantor measure [Formula: see text] except the contraction ratio [Formula: see text] being some algebraic numbers called odd-trinomial number. When [Formula: see text] is an odd-trinomial number, we provide an exponential and polynomial estimations of the upper bound of the spectral number related to the algebraic degree of [Formula: see text]. Some examples on odd-trinomial number via generalized Fibonacci numbers are provided such that the spectral number of them can be determined. Our study involves techniques from polynomial theory, especially the decomposition theory on trinomial.