3-Fibonacci Polynomials in The Family of Fibonacci Numbers

Bell polynomials and Fibonacci numbers

Fibonacci numbers and orthogonal polynomials

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...

Energy of Graphs, Matroids and Fibonacci Numbers

The energy E(G) of a graph G is the sum of the absolute values of the eigenvalues of G. In this article we consider the problem whether generalized Fibonacci constants $varphi_n$ $(ngeq 2)$ can be the energy of graphs. We show that $varphi_n$ cannot be the energy of graphs. Also we prove that all natural powers of $varphi_{2n}$ cannot be the energy of a matroid.

Distribution of the Fibonacci numbers modulo 3

For any modulus m ≥ 2 and residue b (mod m) (we always assume 1 ≤ b ≤ m), denote by ν(m, b) the frequency of b as a residue in one period of the sequence {Fn (mod m)}. It was proved that ν(5k, b) = 4 for each b (mod 5k) and each k ≥ 1 by Niederreiter in 1972 [7]. Jacobson determined ν(2k, b) for k ≥ 1 and ν(2k5j , b) for k ≥ 5 and j ≥ 0 in 1992 [6]. Some other results in this area can be found ...

-̂fibonacci Polynomials

Let MC be the monoid of all Morse code sequences of dots a(:=®) and dashes b(: = -) with respect to concatenation. MC consists of all words in a and b. Let P be the algebra of all polynomials HveMCK ^h r e a l coefficients. We are interested in: a) polynomials in P which we call abstract Fibonacci polynomials. They are defined by the recursion Fn(a, b) = aF^a, b) + bFn_2(a, b) with initial valu...