Related papers: Fibonacci Numbers and Identities II

We use rook placements to prove Spivey's Bell number formula and other identities related to it, in particular, some convolution identities involving Stirling numbers and relations involving Bell numbers. To cover as many special cases as…

We give enumerations of various families of restricted permutations involving the Fibonacci numbers or k-generalized Fibonacci numbers.

Using a probabilistic approach, we derive some interesting combinatorial identities involving gamma and beta functions. These results generalize certain well-known combinatorial identities involving binomial coefficients and special…

We generalize the "motivated proof" of the Rogers-Ramanujan identities given by Andrews and Baxter to provide an analogous "motivated proof" of Gordon's generalization of the Rogers-Ramanujan identities. Our main purpose is to provide…

This paper considers the properties of Tribonacci numbers on identities, matrices, and determinants. In the first front part, we obtain several symmetric identities of Tribonacci numbers by a matrix-based approach and binomial inversion…

We express the N\"{o}rlund polynomials in terms of the second-order Eulerian numbers. Based on this expression, we derive several identities related to the Bernoulli numbers. In particular, we present a short proof of the problem raised by…

We provide bijective proofs of two classic identities that are very simple to prove using generating functions, but surprisingly difficult to prove combinatorially. The problem of finding a bijective proof for the first identity was first…

We provide a formula for the $n^{th}$ term of the $k$-generalized Fibonacci-like number sequence using the $k$-generalized Fibonacci number or $k$-nacci number, and by utilizing the newly derived formula, we show that the limit of the ratio…

We consider $m$-th order linear recurrences that can be thought of as generalizations of the Lucas sequence. We exploit some interplay with matrices that again can be considered generalizations of the Fibonacci matrix. We introduce the…

It is shown how some of the recent results of de Souza et al. [1] can be generalized to describe Hamiltonians whose eigenvalues are given as k-generalized Fibonacci numbers. Here k is an arbitrary integer and the cases considered by de…

We give a survey of some known and some new results about factors of different sorts of $q-$Fibonacci numbers.

We give a simple proof of a recent result by J. Schleischitz dealing with a counterexample to the uniform Littlewood conjecture. Our construction is based on simple properties of Fibonacci numbers.

Starting with a known polynomial identity, we derive two generalizations of Touchard's identity concerning Catalan numbers; one obtained using the Beta function and the other via a connection with Stirling numbers of the second kind. We…

In this paper, we obtain a generalization of an identity due to Carlitz on Bernoulli polynomials. Then we use this generalized formula to derive two symmetric identities which reduce to some known identities on Bernoulli polynomials and…

We present an identity which can be regarded as a Pfaffian-Hafnian analogue of Borchardt's identity and as a generalization of Schur's identity. We give a proof using the complex analysis.

We prove a generalization of the Capelli identity. As an application we obtain an isomorphism of the Bethe subalgebras actions under the (gl(N),gl(M)) duality.

In this paper, we find all integers $c$ having at least two representations as a difference between a Fibonacci number and a Tribonacci number.

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 give new proofs of MacMahon and Russell's modulo 6 identities using the method of weighted words. We also present a new refinement of MacMahon's identity, some related finite sum identities, and a companion partition theorem to sequence…

Ozeki and Prodinger showed that the odd power sum of the first several consecutive Fibonacci numbers of even order is equal to a polynomial evaluated at certain Fibonacci number of odd order. We prove that this polynomial and its derivative…