Orthogonal Polynomials on the Unit Circle with Fibonacci Verblunsky Coefficients, II. Applications

We consider CMV matrices with Verblunsky coefficients determined in an appropriate way by the Fibonacci sequence and present two applications of the spectral theory of such matrices to problems in mathematical physics. In our first application we estimate the spreading rates of quantum walks on the line with time-independent coins following the Fibonacci sequence. The estimates we obtain are explicit in terms of the parameters of the system. In our second application, we establish a connection between the classical nearest neighbor Ising model on the one-dimensional lattice in the complex magnetic field regime, and CMV operators. In particular, given a sequence of nearest-neighbor interaction couplings, we construct a sequence of Verblunsky coefficients, such that the support of the Lee-Yang zeros of the partition function for the Ising model in the thermodynamic limit coincides with the essential spectrum of the CMV matrix with the constructed Verblunsky coefficients. Under certain technical conditions, we also show that the zeros distribution measure coincides with the density of states measure for the CMV matrix.Comment: 23 page

ABC Implies There are Infinitely Many non-Fibonacci-Wieferich Primes - An Application of ABC Conjecture over Number Fields

In this paper, we define $X$-base Fibonacci-Wieferich prime which is a generalized Wieferich prime where $X$ is a finite set of algebraic numbers. We are going to show that there are infinitely many non-$X$-base Fibonacci-Wieferich primes assuming the $abc$-conjecture of Masser-Oesterl\'{e}-Szpiro for number fields. We also provide a new conjecture concerning the rank of free part of abelian group generated by all elements in $X$, and we will use the arithmetic point of view and geometric point of view to give heuristic.Comment: 16 page

The unifying formula for all Tribonacci-type octonions sequences and their properties

Various families of octonion number sequences (such as Fibonacci octonion, Pell octonion and Jacobsthal octonion) have been established by a number of authors in many different ways. In addition, formulas and identities involving these number sequences have been presented. In this paper, we aim at establishing new classes of octonion numbers associated with the generalized Tribonacci numbers. We introduce the Tribonacci and generalized Tribonacci octonions (such as Narayana octonion, Padovan octonion and third-order Jacobsthal octonion) and give some of their properties. We derive the relations between generalized Tribonacci numbers and Tribonacci octonions.Comment: arXiv admin note: text overlap with arXiv:1710.0060

On the existence of ratio limits of weighted nn-generalized Fibonacci sequences with arbitrary initial conditions

We study ratio limits of the consecutive terms of weighted $n$-generalized Fibonacci sequences generated from arbitrary complex initial conditions by linear recurrences with arbitrary complex weights. We prove that if the characteristic polynomial of such a linear recurrence is asymptotically simple, then the ratio limit exists for any sequence generated from arbitrary nontrivial initial conditions and it is equal to the unique zero of the characteristic polynomial.Comment: 3 page

A new Fibonacci identity and its associated summation identities

We derive a new Fibonacci identity. This single identity subsumes important known identities such as those of Catalan, Ruggles, Halton and others, as well as standard general identities found in the books by Vajda, Koshy and others. We also derive several binomial and ordinary summation identities arising from this identity; in particular we obtain a generalization of Halton's general Fibonacci identity.Comment: 12 pages, no figure

Hyperbolic k-Fibonacci Quaternions

In this paper, hyperbolic k-Fibonacci quaternions are defined. Also, some algebraic properties of hyperbolic k-Fibonacci quaternions which are connected with hyperbolic numbers and k-Fibonacci numbers are investigated. Furthermore, D'Ocagne's identity, the Honsberger identity, Binet's formula, Cassini's identity and Catalan's identity for these quaternions are given

Multidimensional Fibonacci Coding

Fibonacci codes are self-synchronizing variable-length codes that are proven useful for their robustness and compression capability. Asymptotically, these codes provide better compression efficiency as the order of the underlying Fibonacci sequence increases, but at the price of the increased suffix length. We propose a circumvention to this problem by introducing higher-dimensional Fibonacci codes for integer vectors. In the process, we provide extensive theoretical background and generalize the theorem of Zeckendorf to higher order. As thus, our work unify several variations of Zeckendorf's theorem while also providing new grounds for its legitimacy

The Bi-periodic Fibonacci Octonions

In this paper, by using bi-periodic Fibonacci numbers, we introduce the bi-periodic Fibonacci octonions. After that, we derive the generating function of these octonions as well as investigated some properties over them. Also, as another main result of this paper, we give the summations for bi-periodic Fibonacci octonions

Nonclassical properties of electronic states of aperiodic chains in a homogeneous electric field

The electronic energy levels of one-dimensional aperiodic systems driven by a homogeneous electric field are studied by means of a phase space description based on the Wigner distribution function. The formulation provides physical insight into the quantum nature of the electronic states for the aperiodic systems generated by the Fibonacci and Thue-Morse sequences. The nonclassical parameter for electronic states is studied as a function of the magnitude of homogeneous electric field to achieve the main result of this work which is to prove that the nonclassical properties of the electronic states in the aperiodic systems determine the transition probability between electronic states in the region of anticrossings. The localisation properties of electronic states and the uncertainty product of momentum and position variables are also calculated as functions of the electric field.Comment: 8 pages, 11 figure

Simson Identity of Generalized m-step Fibonacci Numbers

One of the best known and oldest identities for the Fibonacci sequence $F_n$ is $F_{n+1}F_{n-1}-F_{n}^2=(-1)^n$ which was derived first by R. Simson in 1753 and it is now called as Simson or Cassini Identity. In this paper, we generalize this result to generalized m-step Fibonacci numbers and give an attractive formula. Furthermore, we present some Simson's identities of particular generalized m-step Fibonacci sequences