Related papers: Three new classes of binomial Fibonacci sums
Consider the weight $\lambda$ which is the sum of all simple roots of a simple Lie algebra. Using Kostant's weight multiplicity formula we describe and enumerate the contributing terms to the multiplicity of the zero weight in the…
Roman logarithmic binomial formula analogue has been found . It is presented here also for the case of fibonomial coefficients which recently have been given a combinatorial interpretation by the present author.
In this note we present examples of cumulative connection constants included new fibonomial ones. All examples posses combinatorial interpretation.
We introduce the notion of Fibonacci and Lucas derivations of the polynomial algebras and prove that any element of kernel of the derivations defines a polynomial identity for the Fibonacci and Lucas polynomials. Also, we prove that any…
We study a new class of polyominoes, called $p$-Fibonacci polyominoes, defined using $p$-Fibonacci words. We enumerate these polyominoes by applying generating functions to capture geometric parameters such as area, semi-perimeter, and the…
In this paper we state some conjectures about q-Fibonacci polynomials which for q=1 reduce to well-known results about Fibonacci numbers and Fibonacci polynomials.
We show that the product of two consecutive Fibonacci (respectively Lucas) numbers is divisible by the sum of their indices if this sum is a prime number different from 5 and in the form (4r+1)(respectively (4r+3)).
We give a new identity involving Bernoulli polynomials and combinatorial numbers. This provides, in particular, a Faulhaber-like formula for sums of the form $1^m (n-1)^m + 2^m (n-2)^m + \cdots + (n-1)^m 1^m$ for positive integers $m$ and…
We find various series that involves the central binomial coefficients $\binom{2n}{n}$, harmonic numbers and Fibonacci Numbers.\\ Contrary to the traditional hypergeometric function $_pF_q$ approach, our method utilizes a straightforward…
In this short note, we prove a conjecture recently posed by Alekseyev, Amdeberhan, Shallit, and Vukusic on the 3-adic valuation of a cubic binomial sum.
The class of involutive bisemilattices plays the role of the algebraic counterpart of paraconsistent weak Kleene logic. Involutive bisemilattices can be represented as Plonka sums of Boolean algebras, that is semilattice direct systems of…
We introduce a subclass of linear recurrence sequences which we call poly-rational sequences because they are denoted by rational expressions closed under sum and product. We show that this class is robust by giving several…
The focus of this paper is the study of generalized Fibonacci polynomials and Fibonomial coefficients. The former are polynomials {n} in variables s and t given by {0} = 0, {1} = 1, and {n} = s{n-1}+t{n-2} for n ge 2. The latter are defined…
In \cite{Ka}, the authors obtained a method for deriving special matrices, whose powers are related to Fibonacci and Lucas numbers. In the study, it has been developed a method for deriving special matrices of $3\times 3$ dimensions, whose…
Up to our knowledge -since about 126 years we were lacking of classical type combinatorial interpretation of Fibonomial coefficients as it was Lukas \cite{1} - to our knowledge -who was the first who had defined Finonomial coefficients and…
Let $(F_n)_{n\geq 0}$ and $(L_n)_{n\geq 0}$ be the Fibonacci and Lucas sequences, respectively. In this paper we determine all Fibonacci numbers which are mixed concatenations of a Fibonacci and a Lucas numbers. By mixed concatenations of $…
This note is dedicated to Professor Gould. The aim is to show how the identities in his book "Combinatorial Identities" can be used to obtain identities for Fibonacci and Lucas polynomials. In turn these identities allow to derive a wealth…
We give some results and conjectures about recurrence relations for certain sequences of binomial sums.
In this paper, we define Tribonacci and Tribonacci-Lucas matrix sequences and investigate their properties.
We perform certain alternating binomial summations with parameters that occur in the analysis of algorithms. A combination of integral and special function and special number representations is used. The results are sufficiently general to…