We study the equation Fb = Fx(1)+F−x(3)+F−x(4)+ · · ·+F−x(m) with m ≥ 3, 0 < x(1) < b, and 0 < x(3) < x(4) < · · · < x(m). This equation naturally arises in the generalization of several problems that have appeared in The Fibonacci Quarterly in the problem sections. This equation also has intrinsic interest in its own right. The main theorem the Accident theorem–states, that under very mild con...

The discrete Schrödinger equation with a quasiperiodic dichotomous potential specified by the Fibonacci sequence is known to have a singular continuous eigenvalue spectrum with all states being critically localized. This equation can be transformed into a quasiperiodic skew product dynamical system. In this iterative mapping which is entirely equivalent to the Schrödinger problem, critically lo...

In this article we develop a method of solving general one-dimensional Linear Quadratic Regulator (LQR) problems in optimal control theory, using a generalized form of Fibonacci numbers. We find the solution R (k) of the corresponding discrete-time Riccati equation in terms of ratios of generalized Fibonacci numbers. An explicit Binet type formula for R (k) is also found, removing the need for ...

Fibonacci numbers and difference equations show up in many counting problems. Zeckendorf showed how to represent natural numbers in “binary” Fibonacci bases. Capocelli counted the number of 0 bits and 1 bits in such representation. Here we use the theory of difference equations to try to provide proofs for Capocelli’s claims. We also investigate generalization of the Fibonacci difference equati...

In this study, the Fibonacci collocation method based on the Fibonacci polynomials are presented to solve for the fractional diffusion equations with variable coefficients. The fractional derivatives are described in the Caputo sense. This method is derived by expanding the approximate solution with Fibonacci polynomials. Using this method of the fractional derivative this equation can be reduc...

in this paper, we investigate the generalizedhyers--ulam stability of the functional equation

Let us begin by defining a generalized Fibonacci sequence (gn) with all gn in some abelian group as a sequence that satisfies the recurrence gn = gn−1 + gn−2 as n ranges over Z. The Fibonacci sequence (Fn) is the generalized Fibonacci sequence with integer values defined by F0 = 0 and F1 = 1. Recall also the Binet formula: for any integer n, Fn = (α − β)/ √ 5, where α = (1 + √ 5)/2 and β = (1− ...

Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric have been studied in the literature with help generating functions their functional equations. In this paper, using (p,q)–Fibonacci (p,q)–Lucas Changhee numbers, we define (p,q)–Fibonacci–Changhee polynomials (p,q)–Lucas–Changhee respectively. We obtain some important identities relations these new...

‎hensel [k‎. ‎hensel‎, ‎deutsch‎. ‎math‎. ‎verein‎, ‎{6} (1897), ‎83-88.] discovered the $p$-adic number as a‎ ‎number theoretical analogue of power series in complex analysis‎. ‎fix ‎a prime number $p$‎. ‎for any nonzero rational number $x$‎, ‎there‎ ‎exists a unique integer $n_x inmathbb{z}$ such that $x = ‎frac{a}{b}p^{n_x}$‎, ‎where $a$ and $b$ are integers not divisible by ‎$p$‎. ‎then $|x...

(a) For our first diagram, let’s consider the difference equation used to compute Fibonacci numbers: x(t) = x(t − 1)+ x(t − 2). By looking at this equation, we can see that the output at time t is the sum of the output of the previous two time steps. Therefore, we can write this equation as a block diagram using delay elements and a summation element. Draw the block diagram. (b) Now, let’s cons...