Related papers: Generalized Fibonacci Sequences and Binet-Fibonacc…
In this article, we will discover some new generalized identity regarding continued fractions. We will connect the results to Fibonacci numbers and Lucas numbers. For all the proof, we will use induction.
We considered the properties of generalized Fibonacci and Lucas numbers class. The analogues of well-known Fibonacci identities for generalized numbers are obtained. We gained a new identity of product convolution of generalized Fibonacci…
The General Curve Lemma is a tool of Infinite-Dimensional Analysis, which enables refined studies of differentiability properties of mappings between real locally convex spaces. In this article, we generalize the General Curve Lemma in two…
In this article, we give a formula for the generalization of the binomial coefficient to the complex numbers as a linear combination of $\sinc$ functions. We then give a general formula to compute the integral on the real line of the…
We establish new product identities involving the $q$-analogue of the Fibonacci numbers. We show that the identities lead to alternate expressions of generating functions for close-packed dimers on non-orientable surfaces.
Let (F_n)_{n} be the classical Fibonacci sequence. It is well-known that it satisfies F_{n}^2 + F_{n+1}^2 = F_{2n+1}. In this study, we explore generalizations of this Diophantine equation in several directions. First, we solve the…
In this paper we consider particular generalized compositions of a natural number with a given number of parts. Its number is a weighted polynomial coefficient. The number of all generalized compositions of a natural number is a weighted…
We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.
An integer sequence is called realizable if it is the count of periodic points of some map. The Fibonacci sequence $(F_n)$ does not have this property, and the Fibonacci sequence sampled along the squares $(F_{n^2})$ also does not have this…
Based on Burnside's parametrization of the algebraic curve $y^2=x^5-x$ we provide remaining attributes of its uniformization: Fuchsian equations and their solutions, accessory parameters, monodromies, conformal maps, fundamental polygons,…
This paper establishes a generalized relationship between the arc length of sinusoidal spirals \(r^n=\cos(n\theta)\) and the area of generalized Lam\'e curves defined by \(x^{2n}+y^{2n}=1\). Building on our previous work connecting the…
For an arbitrary homogeneous linear recurrence sequence of order d with constant coefficients, we derive recurrence relations for all subsequences with indices in arithmetic progression. The coefficients of these recurrences are given…
Vector fields with components which are generalized zero-forms are constructed. Inner products with generalized forms, Lie derivatives and Lie brackets are computed. The results are shown to generalize previously reported results for…
We present here some new identities for generalizations of Fibonacci and Lucas numbers by combinatorially interpreting these numbers in terms of numbers of certain tilings of a $1 \times m$ board. As a consequence, some new interesting…
Based on the structure of Fibonacci sequence, we give a new proof for the irrationality exponents of the Fibonacci real numbers. Moreover, we obtain all the irrationality exponents of the real numbers corresponding to the differences of…
It is known that the greatest common divisor of two Fibonacci numbers is again a Fibonacci number. It is called the strong divisibility property. However, this property does not hold for every second order sequence. In this paper we study…
A second order polynomial sequence is of \emph{Fibonacci-type} (\emph{Lucas-type}) if its Binet formula has a structure similar to that for Fibonacci (Lucas) numbers. Known examples of these type of sequences are: Fibonacci polynomials,…
The Fibonacci cube $\Gamma_n$ is the subgraph of the hypercube $Q_n$ induced by vertices with no consecutive $1$s. Recently Jianxin Wei and Yujun Yang introduced a one parameter generalization, Fibonacci $p$-cubes $\Gamma_n^p$, which are…
A sequence of nonzero integers $f = (f_1, f_2, \dots)$ is ``binomid'' if every $f$-binomid coefficient $\left[\! \begin{array}{c} n \\ k \end{array}\! \right]_f$ is an integer. Those terms are the generalized binomial coefficients: \[…
The close relationship among the polynomial functions and Fibonacci numerical sequences is shown in this paper. These numerical sequences are defined by the recurrence equation $x_{k + n} = \displaystyle\sum_{j = 0}^{n-1}\alpha_j x_{k +…