A Note on Extended Binomial Coefficients
Thorsten Neuschel
我们研究了扩展的二项式系数的分布，导出了一个完整的一致误差项的渐近展开。我们从局部中心极限定理中得到展开式，并将所有系数明确表示为Hermite多项式和Bernoulli数的和。
We study the distribution of the extended binomial coefficients by deriving a complete asymptotic expansion with uniform error terms. We obtain the expansion from a local central limit theorem and we state all coefficients explicitly as sums of Hermite polynomials and Bernoulli numbers.
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A note on Lucas L binomials and Ward Horadam binomials recurrences
本文给出了Ward Horadam函数列指定的H（x）二项式递推公式，这是几十年来大多数被认为是多项式的情况。
A note on V-binomials recurrence for Lucas companion to $U_n$ sequence $V_n$
交付Lucas伴随$U_n$序列$V_n$的V-二项式复发
A note on some binomial sums
对E. Brietzke我们给出了简单的证明的递归关系的一些序列的二项式和以前已经获得了其他更复杂的方法。
Notes on q-analogues of a binomial congruence of Glaisher
安德鲁斯曾经给了$q$ -类似物的二项式同余Glaucine，他建议完善的$q$ -类似物。本文给出了满足Andrews要求的参数。
Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II
我们在这里提供第二个新的$\textit{H(x)}-binomials'$递推公式，其中$H(x)-binomials' $阵列是由$Ward-Horadam$序列的功能，在主要考虑的情况下，选择多项式指定。其次，我们提供了一个审查选定的相关组合解释的广义二项式系数。然后，我们还提出了一种转移的解释$p,q-binomial $系数到$q-binomial$系数的解释，从而使我们回到$Gy{ö}rgy Pólya $和唐纳德欧文克努特相关的调查几十年前。
Note on a double binomial sum relevant to the Hadamard maximal determinant problem
我们证明了一个双二项和等式，它与大多数二项和等式的不同之处在于被加数涉及绝对值函数。该恒等式是有趣的，因为它可以用于证明Hadamard极大行列式问题的下界。我们的身份证明使用了两个变量的方法的伸缩总和的变体。
Verification of Binomial theorem and Chu-Vandermonde convolution by the finite difference method
本文证明了二项式定理和Chu-Vandermonde卷积都可以用有限差分法验证。
We derive asymptotic formulas for central extended binomial coefficients, which are generalizations of binomial coefficients. To do so, we relate the exact distribution of the sum of independent discrete uniform random variables to the asymptotic distribution, obtained from the Central Limit Theorem and a local limit variant.
Short note on the convolution of binomial coefficients
我们知道[鲁伊·杜阿尔特和安东尼奥·格德斯·德·奥利维拉，旧身份的新发展，手稿arXiv：1203.5424，提交。即，对于每个非负整数$n,i,j$和对于每个实数$\ell$，已知为$4^n$的$$ \sum_{i+j=n} \binom{2i-\ell}{i} \binom{2j+\ell}{j} = \sum_{i+j=n}\binom{2i}{i} \binom{2j}{j}, $$。我们扩展了这一结果证明，的确，$$ \sum_{i+j=n} \binom{ai+k-\ell}{i} \binom{aj+\ell}{j} = \sum_{i+j=n} \binom{ai+k}{i} \binom{aj}{j} $$的每一个整数$a$和每一个真正的$k$，并提出了新的表达式为这个值。
Certain Binomial Sums with recursive coefficients
本文建立了一些包含系数满足三阶线性递归关系的二项式和的恒等式。作为一个结果，特别是，我们得到了一般形式的早期身份涉及二项式系数和斐波那契型序列。
In this note, we confirm two conjectural supercongruences on double sums of binomial coefficients due to El Bachraoui.
In 1992, Loeb considered a natural extension of the binomial coefficients to negative entries and gave a combinatorial interpretation in terms of hybrid sets. He showed that many of the fundamental properties of binomial coefficients continue to hold in this extended setting. Recently, Formichella and Straub showed that these results can be extended to the $q$-binomial coefficients with arbitrary integer values and extended the work of Loeb further by examining arithmetic properties of the $q$-binomial coefficients. In this paper, we give an alternative combinatorial interpretation in terms of…
The "Shape" of q-Binomial Coefficients
本文以说明性札记形式梳理q-二项式系数的渐近行为与定性性质，并用全新图表呈现其少为人知的整体“形状”。
A Class of Binomial Permutation Polynomials
本文给出了一类二项式是置换多项式的一个判别准则。由此得到了许多类二项置换多项式和单项完全置换多项式。这些单项式中的指数是Niho型的。
Note on Doron Zeilberger's paper: ${5 \choose 2}$ proofs that ${n \choose k} \leq {n \choose {k+1}}$ if $k < n/2$
我们提出了二项式系数单峰性的另一个代数证明。
The Binomial Coefficient for Negative Arguments
根据Γ函数的二项式系数的定义也允许非整数参数。对于非负整数参数，Γ函数可简化为fascal，从而得到著名的杨辉三角形。使用Γ函数的对称公式，这个定义被扩展到负整数参数，使得二项式系数的对称性对所有整数参数都有效。这个定义与其他一些身份和二项式定理的协议进行了调查。
Proof of a conjecture related to divisibility properties of binomial coefficients
设$a,b$和$n$是正整数，其中$a>b$。本文证明了$$(2bn+1)(2bn+3){2bn \choose bn}\bigg|3(a-b)(3a-b){2an \choose an}{an\choose bn}.$$，这证实了Amdeberhan和Moll最近的一个猜想.
Sums of products of binomial coefficients mod 2 and run length transforms of sequences
该研究推导了二项式系数乘积之和模2的递推关系，证明其生成序列为已知基本序列的游程长度变换，并给出正整数、斐波那契数等序列游程长度变换的公式。
Asymptotic products of binomial and multinomial coefficients revisited
该研究重审二项式与多项式系数的渐近乘积，确定其渐近常数与公式，推导行系数的极限情形，明确常数区间与精确表达式，修正了Hirschhorn的相关结果。
Loeb showed that a natural extension of the usual binomial coefficient to negative (integer) entries continues to satisfy many of the fundamental properties. In particular, he gave a uniform binomial theorem as well as a combinatorial interpretation in terms of choosing subsets of sets with a negative number of elements. We show that all of this can be extended to the case of Gaussian binomial coefficients. Moreover, we demonstrate that several of the well-known arithmetic properties of binomial coefficients also hold in the case of negative entries. In particular, we show that Lucas' Theorem …
In this paper, we use the generalized q-polynomials with double q-binomial coefficients and homogeneous q-operators [J. Difference Equ. Appl. 20 (2014), 837--851.] to construct q-difference equations with seven variables, which generalize recent works of Jia et al [Symmetry 2021, 13, 1222.]. In addition, we derive Rogers formulas, extended Rogers formulas and Srivastava--Agarwal type bilinear generating functions for generalized q-polynomials, which generalize generating functions for Cigler's polynomials [J. Difference Equ. Appl. 24 (2018), 479--502.]. Finally, we also derive mixed generating…
Combinatorial interpretations of binomial coefficient analogues related to Lucas sequences
Let s and t be variables. Define polynomials {n} in s, t by {0}=0, {1}=1, and {n}=s{n-1}+t{n-2} for n >= 2. If s, t are integers then the corresponding sequence of integers is called a Lucas sequence. Define an analogue of the binomial coefficients by C{n,k}={n}!/({k}!{n-k}!) where {n}!={1}{2}...{n}. It is easy to see that C{n,k} is a polynomial in s and t. The purpose of this note is to give two combinatorial interpretations for this polynomial in terms of statistics on integer partitions inside a k by n-k rectangle. When s=t=1 we obtain combinatorial interpretations of the fibonomial coeffic…
Every binomial coefficient aficionado knows that the greatest common divisor of the binomial coefficients $\binom n1,\binom n2,\dots,\binom n{n-1}$ equals $p$ if $n=p^i$ for some $i>0$ and equals 1 otherwise. It is less well known that the greatest common divisor of the binomial coefficients $\binom{2n}2,\binom{2n}4,\dots,\binom{2n}{2n-2}$ equals (a certain power of 2 times) the product of all odd primes $p$ such that $2n=p^i+p^j$ for some $0\le i\le j$. This note gives a concise proof of a tidy generalization of these facts.
This work presents a study of perturbations of symmetric Boolean functions. In particular, it establishes a connection between exponential sums of these perturbations and Diophantine equations of the form $$ \sum_{l=0}^n \binom{n}{l} x_l=0,$$ where $x_j$ belongs to some fixed bounded subset $Γ$ of $\mathbb{Z}$. The concepts of trivially balanced symmetric Boolean function and sporadic balanced Boolean function are extended to this type of perturbations. An observation made by Canteaut and Videau for symmetric Boolean functions of fixed degree is extended. To be specific, it is proved that, exc…
Some Identities Involving Three Kinds of Counting Numbers
本文给出了几个涉及二项式系数和两类Stirling数的恒等式。
A note on two identities arising from enumeration of convex polyominoes
出于一些二项式系数的身份遇到我们的方法来枚举凸多项式，我们证明了一些更一般的同一类型的身份，其中之一原来是一个奇怪的评价${}_3F_2$的Gessel和斯坦顿。
A q-rious positivity
对于整数$0\le m\le n$，已知$q$ -二项式系数$\qbinom{n}{m}=\prod_{i=1}^m(1-q^{n-m+i})/(1-q^i)$是具有非负整数系数的多项式.这很容易从$q$ -二项式定理，或$\qbinom{n}{m}$的许多组合解释。在本文中，我们猜想一个算术动机的推广的非负性质的产品的比率$q$ -因子，碰巧是多项式。
We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word. Similarly to the Sierpiński gasket that can be built as the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from Pascal triangle modulo $2$, we describe and study the first properties of the subset of $[0, 1] \times [0, 1]$ associated with this extended Pascal triangle modulo a prime $p$.
We give a new $q$-$(1+q)$-analogue of the Gaussian coefficient, also known as the $q$-binomial which, like the original $q$-binomial $\genfrac{[}{]}{0pt}{}{n}{k}_{q}$, is symmetric in $k$ and $n-k$. We show this $q$-$(1+q)$-binomial is more compact than the one discovered by Fu, Reiner, Stanton and Thiem. Underlying our $q$-$(1+q)$-analogue is a Boolean algebra decomposition of an associated poset. These ideas are extended to the Birkhoff transform of any finite poset. We end with a discussion of higher analogues of the $q$-binomial.
Zeilberger's KOH theorem and the strict unimodality of q-binomial coefficients
最近一个不错的结果，由于我。Pak和G. Panova是$q$ -二项式系数$\binom{a+b}{b}_q$的严格单峰（参见\cite{PP}和\cite{PP2}，以获得其定理的稍微修改版本）。由于他们的证明使用了表示论和克罗内克系数，作者还要求使用蔡尔伯格的KOH定理进行论证。本文给出了这样一个证明。然后，作为我们的方法的进一步应用，我们还提供了一个简短的证明，他们的猜想，$\binom{a+b}{b}_q$的连续系数之间的差异可以得到任意大，当我们假设$b$是固定的，$a$是足够大的。
An extension of Lucas Theorem
本文给出计算模素数二项式系数的同余判据的初等证明，并据此推广Lucas定理，应用于推导二项式系数和的恒等式与同余式。
Some Elementary Congruences for the Number of Weighted Integer Compositions
一个非负整数$n$的整数复合是一个非负整数的元组$(π_1,\ldots,π_k)$，其和是$n$ ; $π_i$被称为复合的部分。对于固定数量的$k$部件，$f$加权整数组合（在文献中也称为$f$着色整数组合）的数量由扩展二项式系数$\binom{k}{n}_{f}$给出，其中每个部件尺寸$s$可能出现在$f(s)$不同的颜色中.我们推导出了$\binom{k}{n}_{f}$的几个同余性质，其中大部分类似于普通二项式系数。其中有奇偶性的$\binom{k}{n}_{f}$，巴贝奇的同余，卢卡斯定理等，我们还给出了同余$c_{f}(n)$，数的$f$加权整数组成的任意多个部分，并为扩大二项式系数和。我们关闭我们的结果的应用程序，以总理的标准加权整数组成。
Sums of products of binomial coefficients mod 2 and 2-regular sequences
本文证明了模2二项式系数乘积之和对应的线性递推序列游程长度变换是2-正则序列，从而利用计算机程序 Walnut 复现并扩展了 Wu 的结果，免去了冗长的技术证明。
On a conjecture of Han and Xiong for fractional Gaussian binomial coefficients
本文针对Han和Xiong提出的分数高斯二项式系数猜想，通过证明支持度优势定理解决了大部分情形，将猜想简化为有限单位分数情形，结合计算机计算和Lean中的自动验证，最终验证了所有正有理数$r$及$k\neq200$时猜想成立。
By polynomial (or extended binomial) coefficients, we mean the coefficients in the expansion of integral powers, positive and negative, of the polynomial $1+t +\cdots +t^{m}$; $m\geq 1$ being a fixed integer. We will establish several identities and summation formulæ parallel to those of the usual binomial coefficients.
Introducing q-deformed binomial coefficients of words
本文引入词的二项式系数的q-类似物，推广经典恒等式并揭示词结构信息，同时定义非交换形式幂级数的q-shuffle与q-infiltration乘积，最终推广Eilenberg关于p-群语言的刻画定理。
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。
In this paper, we use two $q$-operators $\mathbb{T}(a,b,c,d,e,yD_x)$ and $\mathbb{E}(a,b,c,d,e,yθ_x)$ to derive two potentially useful generalizations of the $q$-binomial theorem, a set of two extensions of the $q$-Chu-Vandermonde summation formula and two new generalizations of the Andrews-Askey integral by means of the $q$-difference equations. We also briefly describe relevant connections of various special cases and consequences of our main results with a number of known results.
For a prime $p$ and nonnegative integers $j$ and $n$ let $\vartheta_p(j,n)$ be the number of entries in the $n$-th row of Pascal's triangle that are exactly divisible by $p^j$. Moreover, for a finite sequence $w=(w_{r-1}\cdots w_0)\neq (0,\ldots,0)$ in $\{0,\ldots,p-1\}$ we denote by $\lvert n\rvert_w$ the number of times that $w$ appears as a factor (contiguous subsequence) of the base-$p$ expansion $n=(n_{μ-1}\cdots n_0)_p$ of $n$. It follows from the work of Barat and Grabner (Digital functions and distribution of binomial coefficients, J. London Math. Soc. (2) 64(3), 2001), that $\vartheta…
In a recent paper, Jason P. Bell and Jeffrey Shallit introduced the notion of {\em Lie complexity} and proved that the Lie complexity function of an automatic sequence is automatic. In this note, we give more facts concerning Lie complexity and define the extended Lie complexity and the prefix Lie complexity. Further, we prove that some proprieties of Lie complexity also hold for the extended Lie complexity. Particularly, we prove that the extended Lie complexity function and the first-order difference sequence of the prefix Lie complexity function of an automatic sequence are both automatic.
Let $p>3$ be a prime and let $a$ be a positive integer. We show that if $p\equiv1\pmod 4$ or $a>1$ then $$\sum_{k=0}^{\lfloor\frac34p^a\rfloor}\frac{\binom{2k}k^2}{16^k}\equivł(\frac{-1}{p^a}\r)\pmod{p^3}$$ with $(-)$ the Jacobi symbol, which confirms a conjecture of Z.-W. Sun. We also establish the following new congruences: \begin{align*}\sum_{k=0}^{(p-1)/2}\frac{\binom{2k}k\binom{3k}k}{27^k}\equiv&ł(\frac p3\r)\frac{2^p+1}3\pmod{p^2}, \\\sum_{k=0}^{(p-1)/2}\frac{\binom{6k}{3k}\binom{3k}k}{(2k+1)432^k}\equiv&ł(\frac p3\r)\frac{3^p+1}4\pmod{p^2}, \\\sum_{k=0}^{(p-1)/2}\frac{\binom{4k}{2k}\bin…
Generalization of Fibonomial Coefficients
继卢卡斯，然后其他斐波那契人克瓦希涅夫斯基介绍，并已开始在十年前公开调查的整体F-多项式系数，其中包括除其他外二项式，高斯和Fibonomial系数与一个新的统一的组合解释表示在蛛网偏序集的分区和平铺离散超盒。本文研究了一类特殊的T-范数系数子族。本文的主要目的是借助于生成函数发展T-范数系数理论。给出了T-多项式的类二项式定理，并给出了它的一些推论。本文给出了T-多项式系数的一种新的组合解释，并与Konvalina的加权盒对象选择方法进行了比较。一个简短的总结已经知道的性质的T-多项式系数。
In this note the precise minimum number of key comparisons any dual-pivot quickselect algorithm (without sampling) needs on average is determined. The result is in the form of exact as well as asymptotic formulæ of this number of a comparison-optimal algorithm. It turns out that the main terms of these asymptotic expansions coincide with the main terms of the corresponding analysis of the classical quickselect, but still---as this was shown for Yaroslavskiy quickselect---more comparisons are needed in the dual-pivot variant. The results are obtained by solving a second order differential equat…
Polynomial Triangles Revisited
多项式三角形是一个数组，其输入是多项式的整数次幂的系数。虽然多项式系数已经出现在一些作品中，但没有系统的论述。在本文中，我们将填补这一空白。我们描述了这些数组的一些方面，概括了类似的性质的二项式系数。介绍了几种多项式系数枚举的组合模型，包括格路模型、自旋链模型和画图游戏得分模型。几个已知的二项式身份，然后扩展。此外，我们还递归地计算了列序列的生成函数。有趣的推论遵循这些递归关系，如新的公式为斐波那契数和厄米多项式的三项式系数。最后，详细研究了表征多项式系数在极限下的熵密度函数的性质。
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