Related papers: Some Weighted Generalized Fibonacci Number Summati…
In this paper, we introduce relations between binomial sums involving (generalized) Fibonacci and Lucas numbers, and different kinds of binomial coefficients. We also present some relations between sums with two and three binomial…
In this paper, new families of generalized Fibonacci and Lucas numbers are introduced. In addition, we present the recurrence relations and the generating functions of the new families for $k=2$.
We study determinants of matrices whose entries are powers of Fibonacci numbers. We then extend the results to include entries that are powers of generalized Fibonacci numbers defined as a second-order linear recurrence relation. These…
We develop closed form expressions for various finite binomial Fibonacci and Lucas sums depending on the modulo 5 nature of the upper summation limit. Our expressions are inferred from some trigonometric identities.
In this paper, we provide new applications of Fibonacci and Lucas numbers. In some circumstances, we find algebraic structures on some sets defined with these numbers, we generalize Fibonacci and Lucas numbers by using an arbitrary binary…
We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.
Let $F_n(k)$ be the generalized Fibonacci number defined by (with $F_i(k)$ abbreviated to $F_i$): $F_n = F_{n-1} + F_{n-2} + \dots + F_{n-k}$, for $n \geq k$, and the initial values $(F_0,F_1,...,F_{k-1})$. Let $B_n(k,j)$ be $F_n(k)$ with…
In this paper we study the sum $$\sum_{j_1+j_2+...+j_d=n}\prod_{i=1}^d F_{k\cdot j_i},$$ where $d\geq2$ and $k\geq1$.
Some generalized multi-sum Chu-Vandermonde identities are presented and proved, generalizing some known multi-sum Chu-Vandermonde identities from literature and adding some quadratic and cubic examples of these identities. Some other…
By applying the classic telescoping summation formula and its variants to identities involving inverse hyperbolic tangent functions having inverse powers of the golden ratio as arguments and employing subtle properties of the Fibonacci and…
We study formulas expressing Fibonacci numbers as sums over compositions using free submonoids of the free monoid of compositions with parts 1 and 2.
The summation formula within pascalian triangle resulting in the fibonacci sequence is extended to the $q$-binomial coefficients $q$-gaussian triangles.
We consider a class of generalized binomials emerging in fractional calculus. After establishing some general properties, we focus on a particular yet relevant case, for which we provide several ready-for-use combinatorial identities,…
Using elementary methods, we establish old and new relations between binomial coefficients, Fibonacci numbers, Lucas numbers, and more.
In this note, we present a simple summation formula for $k$-bonacci numbers. The derivation consists in obtaining the generating function of such numbers, and noting that its evaluation at a particular value yields a formula generalizing a…
Unlike in the case of Fibonacci and Lucas numbers, there is a paucity of literature dealing with summation identities involving the Padovan and Perrin numbers. In this paper, we derive various summation identities for these numbers,…
We derive closed form expressions for finite and infinite Fibonacci-Lucas sums having products of Fibonacci or Lucas numbers in the denominator of the summand. Our results generalize and extend those obtained by pioneer Brother Alfred…
In this paper, we introduce three new classes of binomial sums involving Fibonacci (Lucas) numbers and weighted binomial coefficients.
The equation commonly known as Sury's identity is a deceptively simple summation formula that connects the Lucas numbers, Fibonacci numbers, and powers of two. Many authors have given extensions and generalizations over the years; in this…
Fibonacci polynomials are generalizations of Fibonacci numbers, so it is natural to consider polynomial versions of the various results for Fibonacci numbers. According to Hong, Pongsriiam, Bulawa, and Lee, the generating function of the…