We define the convolved hðxÞ-Fibonacci polynomials as an extension of the classical con-volved Fibonacci numbers. Then we give some combinatorial formulas involving the hðxÞ-Fibonacci and hðxÞ-Lucas polynomials. Moreover we obtain the convolved hðxÞ-Fibo-nacci polynomials from a family of Hessenberg matrices. Fibonacci numbers and their generalizations have many interesting properties and appli...

We explore the properties of Collapse, a number game closely related to Fibonacci words. In doing so, we fully classify the set of periods (minimal or not) of finite Fibonacci words via careful examination of the Exceptional (sometimes called singular) finite Fibonacci words. Collapse is not a game in the Game Theory sense, but rather in the recreational sense, like the 15-puzzle (the game wher...

In this paper we did a generalization of Hadamard product of Fibonacci Q matrix and Fibonacci Q−n matrix for continuous domain. We obtained Hadamard product of the golden matrices in the terms of the symmetrical hyperbolic Fibonacci functions and investigated some properties of Hadamard product of the golden matrices. Mathematics Subject Classification: Primary 11B25, 11B37, 11B39, Secondary 11C20

We introduce two sets of permutations of {1, 2, . . . , n} whose cardinalities are generalized Fibonacci numbers. Then we introduce the generalized q-Fibonacci polynomials and the generalized q-Fibonacci numbers (of first and second kind) by means of the major index statistic on the introduced sets of permutations.

The Fibonacci secant function is constructed in this paper. Fibonacci tan-sec method is presented to search for the exact traveling wave solutions of nonlinear differentialdifference equations (DDEs). This method is essentially equivalent to the classical hyperbolic method. The discrete mKdV lattice and some other lattice equations are chosen to illustrate the efficiency and effectiveness of th...

We study the discretised segments generated by the iterated Tribonacci substitution and the projections of the integer points on them to some plane. After suitable transformations we get a sequence of finite two-dimensional words which tends to a doubly rotational word on Z. (Without scaling we would get the Rauzy fractal.) As an introduction we start with the corresponding case of the Fibonacc...

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− ...

We study the Fibonacci sets from the point of view of their quality with respect to discrepancy and numerical integration. Let {bn}n=0 be the sequence of Fibonacci numbers. The bn-point Fibonacci set Fn ⊂ [0, 1] is defined as Fn := {(μ/bn, {μbn−1/bn})} μ=1, where {x} is the fractional part of a number x ∈ R. It is known that cubature formulas based on Fibonacci set Fn give optimal rate of error...