关于Fibonacci幂级数的系数

On the duals of the Fibonacci and Catalan-Fibonacci polynomials and Motzkin paths

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.

Zeckendorf expansion, Dirichlet series and infinite series involving the infinite Fibonacci word

Let $β=\frac{1+\sqrt{5}}{2}$, $(a_n)_{n \in \mathbb{N}^+}$ be a non-uniform morphic sequence involving the infinite Fibonacci word and $(δ(n))_{n \in \mathbb{N}^+}$ be a positive sequence such that for all positive integers $n$, $δ(n)=\frac{1}{\sqrt{5}}\sum_{j \geq 0}ε_jβ^{j+2}$ if the unique Zeckendorf expansion of $n$ is $n=\sum_{j \geq 0}ε_jF_{j+2}$ with Fibonacci numbers $F_0,F_1,F_2...$. We define and study some Dirichlet series in the form of $\sum_{n\geq 1}\frac{a_n}{(δ(n))^s}$ and relations between them. Moreover, we compute the values of some infinite series involving the infinite Fib…

$q$-Fibonacci数及其二项式类似物的组合解释

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。

卷积卷积斐波那契数

Convoluted convolved Fibonacci numbers

卷积斐波那契数F_j^（r）定义为（1-z-z^2）^{-r}= \sum_{j>=0} F_{j+1} ^（r）z^j.本文考虑了一些可用卷积斐波那契数表示的相关数.这些数字出现在某个数字理论常数的数值计算中。本文是对变换{1/n} \sum_{d|n} mu（d）f（z^d）^{n/d}的一个案例研究，其中f是任何形式级数，mu是Moebius函数），在题为“形式级数Witt变换”的配套论文中对此进行了研究。

与调和数、斐波那契数及中心二项式系数 $\binom{2n}{n}$ 相关的级数

Series Associated with Harmonic Numbers, Fibonacci Numbers and Central Binomial Coefficients $\binom{2n}{n}$

本文通过直接变换而非传统超几何方法，发现了涉及中心二项式系数、调和数与斐波那契数的多种级数，并给出了 $\zeta(2)$ 的新级数表示。

Balancing polynomials, Fibonacci numbers and some new series for $π$

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.

Fibonacci幂级数的系数

The coefficients of a Fibonacci power series

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。

关于若干涉及二项式系数$\binom{3n}{n}$的级数

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。

On the self-convolution of generalized Fibonacci numbers

We focus on a family of equalities pioneered by Zhang and generalized by Zao and Wang and hence by Mansour which involves self convolution of generalized Fibonacci numbers. We show that all these formulas are nicely stated in only one equation involving a bivariate ordinary generating function and we give also a formula for the coefficients appearing in that context. As a consequence, we give the general forms for the equalities of Zhang, Zao-Wang and Mansour.

Generalized Fibonacci polynomials and Fibonomial coefficients

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…

Some remarks on generalized Fibonacci and Lucas polynomials

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.

A Fibonacci analogue of Stirling numbers

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…

关于双侧广义斐波那契数列的线性空间

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。

Demazure flags, $q$--Fibonacci polynomials and hypergeometric series

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…

Chebyshev Polynomials, Sliding Columns, and the $k$-step Fibonacci Numbers

We give a direct and intuitive proof (via sliding some columns up and down) of the following interesting fact: if we write out the Chebyshev polynomials in a chart and take the sums of coefficients along certain diagonals, we obtain the Fibonaccis, the Tribonaccis, the Tetranaccis, and so on.

Fibonacci迭代中Rule 150元胞自动机活动的时间演化

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。

Three Series for the Generalized Golden Mean

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)).

Heisenberg characters, unitriangular groups, and Fibonacci numbers

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…

预制偏序集` Whitney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）。

台球分拆、斐波那契数列、SIP类与箭图

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。

二次代数中幂的通用恒等式及斐波那契恒等式的矩阵导出

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。

广义斐波那契多项式的零点与正交性

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。