The coefficients of a Fibonacci power series
Federico Ardila
我们给出一个简短的证明，证明了级数（1-x）（1-x^2）（1-x^3）（1-x^5）（1-x^8）（1-x^13）（1-x^21）.等于-1，0或1，并且它们中的大多数等于0。
We give a short proof of the result that all the coefficients of the series (1-x)(1-x^2)(1-x^3)(1-x^5)(1-x^8)(1-x^13)(1-x^21)... are equal to -1, 0, or 1, and most of them are equal to 0.
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We use probability urn models to discover some known and unknown series identities involving Fibonacci numbers.
Some power series extensions of Fibonacci and Lucas polynomials
本研究聚焦形式幂级数领域，探讨可作为偶/奇下标斐波那契、卢卡斯多项式插值的形式幂级数，拓展了相关多项式的幂级数研究方向。
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Series Associated with Harmonic Numbers, Fibonacci Numbers and Central Binomial Coefficients $\binom{2n}{n}$
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We evaluate some types of infinite series with balancing and Lucas-balancing polynomials in closed form. These evaluations will lead to some new curious series for $π$ involving Fibonacci and Lucas numbers. Our findings complement those of Castellanos from 1986 and 1989.
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 by {n choose k} = {n}!/({k}!{n-k}!) where {n}! = {1}{2}...{n}. These quotients are also polynomials in s and t, and specializations give the ordinary binomial coefficients, the Fibonomial coefficients, and the q-binomial coefficients. We present some of their fundamental properties, including a more general recursion for {n}, an analogue of the binomial the…
On the coefficients of a Fibonacci power series
本文给出了形式幂级数（1-x）（1-x^2）（1-x^3）（1-x^5）（1-x^8）（1-x^13）.特别地，我们证明了所有的系数都等于-1，0或1。
On the linear space of the two-sided generalized Fibonacci sequences
本文研究了满足 $k$ 阶递推方程的双侧广义斐波那契数列的线性空间，并给出了分别基于广义二项式系数和广义多项式系数的两类显式公式。
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New Binomial Identities for Fibonacci, Lucas, and Generalized Fibonacci Sequences with Multiple Indices
本文提出新的恒等式，通过卢卡斯数的幂和二项式系数表示斐波那契、卢卡斯和广义斐波那契序列的多重索引项，利用对称多项式（瓦ring公式）应用于经典Binet公式。
We study a family of finite--dimensional representations of the hyperspecial parabolic subalgebra of the twisted affine Lie algebra of type $\tt A_2^{(2)}$. We prove that these modules admit a decreasing filtration whose sections are isomorphic to stable Demazure modules in an integrable highest weight module of sufficiently large level. In particular, we show that any stable level $m'$ Demazure module admits a filtration by level $m$ Demazure modules for all $m\ge m'$. We define the graded and weighted generating functions which encode the multiplicity of a given Demazure module and establish…
In this paper we consider particular generalized compositions of a natural number with a given number of parts. Its number is a weighted polynomial coefficient. The number of all generalized compositions of a natural number is a weighted $r$-generalized Fibonacci number. A relationship between these two numbers will be derived. We shall thus obtain a generalization of the well-known formula connecting Fibonacci numbers with the binomial coefficients.
As is well-known, the ratio of adjacent Fibonacci numbers tends to phi = (1 + sqrt(5))/2, and the ratio of adjacent Tribonacci numbers (where each term is the sum of the three preceding numbers) tends to the real root eta of X^3 - X^2 - X - 1 = 0. Letting alpha(n) denote the corresponding ratio for the generalized Fibonacci numbers, where each term is the sum of the n preceding, we obtain rapidly converging series for alpha(n), 1/alpha(n), and 1/(2-alpha(n)).
Binomial sum relations involving Fibonacci and Lucas numbers
该研究探讨了含（广义）斐波那契数、卢卡斯数与不同类型二项式系数的二项式和的关联，给出多类二项式和关系并重新发现部分已作为问题提案的关系式。
We use the inversion of coefficient arrays to define dual polynomials to the Fibonacci and Catalan-Fibonacci polynomials, and we explore the properties of these new polynomials sequences. Many of the arrays involved are Riordan arrays. Direct links to the counting of Motzkin paths by different statistics emerge.
Time-evolution of the Rule 150 cellular automaton activity from a Fibonacci iteration
单种子细胞规则150自动机的总活动不像其他基本细胞自动机那样遵循一步迭代，而是可以作为两步向量或串迭代来求解，这可以被视为从长度增加的向量序列生成时间序列的斐波那契迭代的推广。这允许比通过模拟整个时空过程或者甚至通过使用闭合表达式更有效地计算总活动时间序列。
Combinatorial Interpretations of $q$-Fibonacci Numbers and Their Binomial Analogues
本文提出一个推广Bergeron方法的q-Fibonacci数新模型，为q-Fibonomial系数的阶梯模型提供非双射证明，并展示了涉及该系数的新恒等式，增强了对相关数学对象的组合理解。
Convoluted convolved Fibonacci numbers
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Quadratic Reduction and Classical Multiple-Index Fibonacci-Lucas Identities
研究二次代数中元素幂的通用恒等式，通过矩阵迹和行列式推导斐波那契数列的二项式展开公式，揭示恒等式源于普遍代数原理。
Starting with some determinants of binomial coefficients which are related to Fibonacci and Lucas polynomials we study similar determinants for some generalizations of these polynomials and their q-analogues.
On some series involving the binomial coefficients $\binom{3n}{n}$
该研究利用简单变换简化了Necdet Batir提出的含二项式系数$\binom{3n}{n}$的级数形式，给出新求值结果，建立其与Fibonacci数、黄金比例的关联，还推导了含该系数倒数的Fibonacci与Lucas级数。
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Billiard Partitions, Fibonacci Sequences, SIP Classes, and Quivers
本文从椭球台球周期轨道中的台球分拆出发，引入B型可分离整数分拆类，研究其基本分拆数与斐波那契数列的关系，并建立其生成级数与对称箭图生成级数及Schröder路径计数的联系。
Consider the Fibonacci numbers defined by setting $F_1=1=F_2$ and $F_n =F_{n-1}+F_{n-2}$ for $n \geq 3$. We let $n_F! = F_1 \cdots F_n$ and $\binom{n}{k}_F = \frac{n_F!}{k_F!(n-k)_F!}$. Let $(x)_{\downarrow_0} = (x)_{\uparrow_0} = 1$ and for $k \geq 1$, $(x)_{\downarrow_k} = x(x-1) \cdots (x-k+1)$ and $(x)_{\uparrow_k} = x(x+1) \cdots (x+k-1)$. Then the Stirling numbers of the first and second kind are the connections coefficients between the usual power basis $\{x^n:n \geq 0\}$ and the falling factorial basis $\{(x)_{\downarrow_n}:n \geq 0\}$ in the polynomial ring $\mathbb{Q}[x]$ and the Lah…
Let $\UT_n(\FF_q)$ denote the group of unipotent $n\times n$ upper triangular matrices over a finite field with $q$ elements. We show that the Heisenberg characters of $\UT_{n+1}(\FF_q)$ are indexed by lattice paths from the origin to the line $x+y=n$ using the steps $(1,0), (1,1), (0,1), (1,1)$, which are labeled in a certain way by nonzero elements of $\FF_q$. In particular, we prove for $n\geq 1$ that the number of Heisenberg characters of $\UT_{n+1}(\FF_q)$ is a polynomial in $q-1$ with nonnegative integer coefficients and degree $n$, whose leading coefficient is the $n$th Fibonacci number…
Generalization of an Identity of Andrews
我们考虑一个由G. G发现的将斐波那契数与帕斯卡三角形联系起来的恒等式。E.安德鲁斯几位作者提供了证明这一身份，他们都相当参与或依赖于复杂的数论参数。不仅给出了一个简单初等的证明，而且证明了该恒等式推广到了Pascal三角形以外的数组。作为应用，我们得到的身份三项式系数和加泰罗尼亚的三角形斐波那契数。
Prefab posets` Whitney numbers
本文在作者最近构造的结构自然有限子集蛛网预制集中引入了自然偏序。然后计算了构成类Stirling数三角阵的第二类子集的Whitney数，并给出了它们的显式表达式.其次，在第二种构造中，我们赋予预构式的集合和以另一种偏序，使它们的钟状数包含Fibonacci三元组序列，以便将著名的二项式牛顿系数与斐波那契数之间的关系推广到它们的亲属的无穷大，其中也有Fibonacci三元组序列和类二项式系数（包括发病率系数）。
In this short note, we establish some identities containing sums of binomials with coefficients satisfying third order linear recursive relations. As a result and in particular, we obtain general forms of earlier identities involving binomial coefficients and Fibonacci type sequences.
Except for Koshy who devotes seven pages to applications of Fibonacci Numbers to electric circuits, most books and the Fibonacci Quarterly have been relatively silent on applications of graphs and electric circuits to Fibonacci numbers. This paper continues a recent trend of papers studying the interplay of graphs, circuits, and Fibonacci numbers by presenting and studying the Circuit Array, an infinite 2-dimensional array whose entries are electric resistances labelling edge values of circuits associated with a family of graphs. The Circuit Array has several features distinguishing it from ot…
In this article we provide with combinatorial proofs of some recent identities due to Sury and McLaughlin. We show that, the solution of a general linear recurrence with constant coefficients can be interpreted as a determinant of a matrix. Also, we derive a determinantal expression of Fibonacci and Lucas numbers. We prove Binets formula for Fibonacci and Lucas numbers in a purely combinatorial way and in course of doing so, we find a determinantal identity, which we think to be new.
Fibonacci numbers and the probability of polygon formation using random length sticks
通过两种互补证明，得出随机长度棍棒无法形成p+1边多边形的概率可表示为涉及p步斐波那契数的项倒数乘积。
Zeros and Orthogonality of generalized Fibonacci polynomials
本文研究了广义斐波那契多项式（GFP）的零点求解与正交性判定，给出了 Favard 定理特例的替代证明，分类了常见 GFP 的正交性，并探讨了正交多项式与 Markov 过程的联系。
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