where the Fn and Ln denote the Fibonacci numbers and Lucas numbers, have been considered in several different ways. Navas [] discussed the analytic continuation of these series. Elsner et al. [] obtained that for any positive distinct integer s, s, s, the numbers ζF (s), ζF (s), and ζF (s) are algebraically independent if and only if at least one of s, s, s is even. Ohtsuka and Na...

While Fibonacci numbers can quite easily be classified as a complete sequence, they have the unusual property that a particular explicitly defined subset of such sequences is complete and also gives unique representations of natural numbers. This can be extended to all integers via the ”nega-Fibonacci” numbers. Further, an unusual binary operation on the natural numbers may be defined and prove...

In this study we define and study the Bivariate Complex Fibonacci and Bivariate Complex Lucas Polynomials. We give generating function, Binet formula, explicit formula and partial derivation of these polynomials. By defining these bivariate polynomials for special cases Fn(x, 1) is the complex Fibonacci polynomials and Fn(1, 1) is the complex Fibonacci numbers. Finally in the last section we gi...

A perfect number is any positive integer that is equal to the sum of its proper divisors. Several years ago, F. Luca showed that the Fibonacci and Lucas numbers contain no perfect numbers. In this paper, we alter the argument given by Luca for the nonexistence of both odd perfect Fibonacci and Lucas numbers, by making use of an 1888 result of C. Servais. We also provide a brief historical accou...

In this paper, and from the definition of a distance between numbers by a recurrence relation, new kinds of k–Fibonacci numbers are obtained. But these sequences differ among themselves not only by the value of the natural number k but also according to the value of a new parameter r involved in the definition of this distance. Finally, various properties of these numbers are studied.

It is a fundamental property of non-letter Lyndon words that they can be expressed as a concatenation of two shorter Lyndon words. This leads to a naive lower bound ⌈log 2 (n)⌉ + 1 for the number of distinct Lyndon factors that a Lyndon word of length n must have, but this bound is not optimal. In this paper we show that a much more accurate lower bound is ⌈logφ(n)⌉ + 1, where φ denotes the gol...

In this paper we establish some sophisticated congruences involving central binomial coefficients and Fibonacci numbers. For example, we show that if p = 2, 5 is a prime then p−1

We prove that the sequence (1/Fn+2)n≥0 of reciprocals of the Fibonacci numbers is a moment sequence of a certain discrete probability, and we identify the orthogonal polynomials as little q-Jacobi polynomials with q = (1− √ 5)/(1+ √ 5). We prove that the corresponding kernel polynomials have integer coefficients, and from this we deduce that the inverse of the corresponding Hankel matrices (1/F...

(*)' Fn = ox. In [1], Cohn solved (*) for c = 1, 2. In [9], we found all solutions of O ) such that <? is prime and either o = 3 (mod 4) or c < 10,000. Harborth & Kemnitz [4] have asked for solutions of (&) for composite values of o. Clearly, it suffices to consider only squarefree values of c. If c < 1000, then c has at most three distinct odd prime factors. Therefore c = kp where p is prime a...