An extension of a curious binomial identity

Curious congruences for Fibonacci numbers

In this paper we establish some sophisticated congruences involving central binomial coefficients and Fibonacci numbers. For example, we show that if $p\not=2,5$ is a prime then $$\sum_{k=0}^{p-1}F_{2k}\binom{2k}{k}=(-1)^{[p/5]}(1-(p/5)) (mod p^2)$$ and $$\sum_{k=0}^{p-1}F_{2k+1}\binom{2k}k=(-1)^{[p/5]}(p/5) (mod p^2).$$ We also obtain similar results for some other second-order recurrences and raise several conjectures.

A combinatorial proof of Sun's "curious" identity

A binomial coefficient identity due to Zhi-Wei Sun is the subject of half a dozen recent papers that prove it by various analytic techniques and establish a generalization. Here we give a simple proof that uses weight-reversing involutions on suitable configurations involving dominos and colorings. With somewhat more work, the method extends to the generalization also.

Flexagons yield a curious Catalan number identity

Hexaflexagons were popularized by the late Martin Gardner in his first Scientific American column in 1956. Oakley and Wisner showed that they can be represented abstractly by certain recursively defined permutations called pats, and deduced that they are counted by the Catalan numbers. Counting pats by number of descents yields the curious identity Sum[1/(2n-2k+1)binom{2n-2k+1}{k}binom{2k}{n-k},k=0..n] = C(n), where only the middle third of the summands are nonzero.

关于n-nacci数、划分和二项系数乘积的一种恒等式

An identity relating $n$-nacci numbers, partitions, and products of binomial coefficients

本文通过研究最终类型和其相关划分，推导出n-nacci数的恒等式，并比较了不同偏序关系。

On Binomial Identities in Arbitrary Bases

We extend the digital binomial identity as given by Nguyen el al. to an identity in an arbitrary base $b$, by introducing the $b-$ary binomial coefficients. We then study the properties of these coefficients such as orthogonality, a link to Lucas' theorem and the corresponding $b-$ary Pascal triangles.

A note on De Concini and Procesi's curious identity

We give a short, case-free and combinatorial proof of de Concini and Procesi's formula for the volume of the simplicial cone spanned by the simple roots of any finite root system. The argument presented here also extends their formula to include the non-crystallographic root systems.

A Combinatorial Identity for the p-Binomial Coefficient Based on Abelian Groups

For non-negative integers $k\leq n$, we prove a combinatorial identity for the $p$-binomial coefficient $\binom{n}{k}_p$ based on abelian p-groups. A purely combinatorial proof of this identity is not known. While proving this identity, for $r\in \mathbb{N}\cup\{0\},s\in \mathbb{N}$ and $p$ a prime, we present a purely combinatorial formula for the number of subgroups of $\mathbb{Z}^s$ of finite index $p^r$ with quotient isomorphic to the finite abelian $p$-group of type $\underlineλ$, which is a partition of $r$ into at most $s$ parts. This purely combinatorial formula is similar to that for …

A curious identity in connection with saddle-point method and Stirling's formula

We prove the curious identity in the sense of formal power series: \[ \int_{-\infty}^{\infty}[y^m] \exp\left(-\frac{t^2}2 +\sum_{j\ge3}\frac{(it)^j}{j!}\, y^{j-2}\right)\mathrm{d} t = \int_{-\infty}^{\infty}[y^m] \exp\left(-\frac{t^2}2+ \sum_{j\ge3}\frac{(it)^j}{j}\, y^{j-2}\right)\mathrm{d} t, \] for $m=0,1,\dots$, where $[y^m]f(y)$ denotes the coefficient of $y^m$ in the Taylor expansion of $f$. The generality of this identity from the perspective of saddle-point method is also examined.

Generalizations of Andrews' curious identities

According to the method of series rearrangement, we establish two generalizations of Andrews' curious $q$-series identity with an extra integer parameter. The limiting cases of them produce two extensions of Andrews' curious $_3F_2(\frac{3}{4})$-series identity with an additional integer parameter. Meanwhile, several related results are also given.

On the least common multiple of $q$-binomial coefficients

In this paper, we prove the following identity $$ \lcm({n\brack 0}_q,{n\brack 1}_q,...,{n\brack n}_q) =\frac{\lcm([1]_q,[2]_q,...,[n+1]_q)}{[n+1]_q}, $$ where ${n\brack k}_q$ denotes the $q$-binomial coefficient and $[n]_q=\frac{1-q^n}{1-q}$. This result is a $q$-analogue of an identity of Farhi [Amer. Math. Monthly, November (2009)].

An identity for the central binomial coefficient

We find the joint distribution of three simple statistics on lattice paths of n upsteps and n downsteps leading to a triple sum identity for the central binomial coefficient {2n}-choose-{n}. We explain why one of the constituent double sums counts the irreducible pairs of compositions considered by Bender et al., and we evaluate some of the other sums.

A binomial coefficient identity associated to a conjecture of Beukers

A combinatorial identity that was needed in Ahlgren and Ono's proof of a certain congruence conjecture of Frits Beukers is stated, and a pointer to its WZ proof is given.

两个好奇的q-超同余式及其推广

Two curious q-supercongruences and their extensions

本文证明了Guo猜想的两类单参数q-超同余式并给出其双参数推广，关键工具为q-二项式定理终止形式与Gasper的Karlsson-Minton型求和公式，同时验证了Wang、Li和Tang的断言。

一个关于优美标号计数的奇妙恒等式

One Curious Identity Counting Graceful Labelings

该研究针对素因子含两个不同素数且指数均为n的正整数a、b，证明完全二部图K_{a,b}的本质不同α-优美标号数等于二项式系数四次方的交错和，建立了图论标号与组合恒等式的关联。

关于带指定加项的奇数部分划分数的一个奇特恒等式的精细视角

A refined view of a curious identity for partitions into odd parts with designated summands

本文针对Andrews等人提出的带指定加项的奇数部分划分数$\mathrm{PDO}(n)$，给出一个奇特恒等式的精细视角，涉及组合构造与恒等式证明。

Note on a double binomial sum relevant to the Hadamard maximal determinant problem

We prove a double binomial sum identity which differs from most binomial sum identities in that the summands involve the absolute value function. The identity is of interest because it can be used in proofs of lower bounds for the Hadamard maximal determinant problem. Our proof of the identity uses a two-variable variant of the method of telescoping sums.

A binomial-coefficient identity arising from the middle discrete series of SU(2,2)

The aim of this paper is to give an elementary proof of certain identities on binomials and state an answer to Remark 8.2 in Takahiro Hayata, Harutaka Koseki, and Takayuki Oda, Matrix coefficients of the middle discrete series of SU(2,2), J. Funct. Anal. 185 (2001), 297-341.

Algebraic identities on q-harmonic numbers and q-binomial coefficients

The aim of this paper is to present a general algebraic identity. Applying this identity, we provide several formulas involving the q-binomial coefficients and the q-harmonic numbers. We also recover some known identities including an algebraic identity of D. Y. Zheng on q-Apéry numbers and we establish the q-analog of Euler's formula. The proposed results may have important applications in the theory of q-supercongruences.

Congruences on sums of $q$-binomial coefficients

We establish a $q$-analogue of Sun--Zhao's congruence on harmonic sums. Based on this $q$-congruence and a $q$-series identity, we prove a congruence conjecture on sums of central $q$-binomial coefficients, which was recently proposed by Guo. We also deduce a $q$-analogue of a congruence due to Apagodu and Zeilberger from Guo's $q$-congruence.

Batır恒等式的应用

Applications of an identity of Batır

本文基于Batır恒等式，推导出与著名数列相关的双和新恒等式，并证明了二项式变换对的双和恒等式。

On some Binomial Coefficient Identities with Applications

We present a different proof of the following identity due to Munarini, which generalizes a curious binomial identity of Simons. \begin{align*} \sum_{k=0}^{n}\binomα{n-k}\binom{β+k}{k}x^k &=\sum_{k=0}^{n}(-1)^{n+k}\binom{β-α+n}{n-k}\binom{β+k}{k}(x+1)^k, \end{align*} where $n$ is a non-negative integer and $α$ and $β$ are complex numbers, which are not negative integers. Our approach is based on a particularly interesting combination of the Taylor theorem and the Wilf-Zeilberger algorithm. We also generalize a combinatorial identity due to Alzer and Kouba, and offer a new binomial sum identity…

All binomial identities are orderable

The main result of this paper is to show that all binomial identities are orderable. This is a natural statement in the combinatorial theory of finite sets, which can also be applied in distributed computing to derive new strong bounds on the round complexity of the weak symmetry breaking task. Furthermore, we introduce the notion of a fundamental binomial identity and find an infinite family of values, other than the prime powers, for which no fundamental binomial identity can exist.

A Strehl Version of Fourth Franel Sequence

We give a combinatorial identity related to the Franel numbers involving the sum of fourth power of binomial coefficients. Furthermore, investigating in J. Mikic's proof of the first Strehl Identity, we provide a combinatorial proof of this identity using the double counting argument.

q-Analogs of some congruences involving Catalan numbers

We provide some variations on the Greene-Krammer's identity which involve q-Catalan numbers. Our method reveals a curious analogy between these new identities and some congruences modulo a prime.

高阶 Mersenne 数序列及相关二项式变换

The sequence of higher order Mersenne numbers and associated binomial transforms

本文提出高阶 Mersenne 整数序列，系统推导其 Binet 公式、恒等式、生成函数、求和关系及矩阵表示，并研究其二项式变换序列。

Chu's 1303 Identity Implies Bombieri's 1990 Norm-Inequality [via an Identity of Beauzamy and Dégot]

The Vandermonde-Chu Binomial Coefficients Identity is shown to imply Bombieri's deep norm inequalities, via identities of Beauzamy-Dégot, and Reznick.

On sums of binomial coefficients and their applications

In this paper we study recurrences concerning the combinatorial sum $[n,r]_m=\sum_{k\equiv r (mod m)}\binom {n}{k}$ and the alternate sum $\sum_{k\equiv r (mod m)}(-1)^{(k-r)/m}\binom{n}{k}$, where m>0, $n\ge 0$ and r are integers. For example, we show that if $n\ge m-1$ then $$\sum_{i=0}^{\lfloor(m-1)/2\rfloor}(-1)^i\binom{m-1-i}i [n-2i,r-i]_m=2^{n-m+1}.$$ We also apply such results to investigate Bernoulli and Euler polynomials. Our approach depends heavily on an identity established by the author [Integers 2(2002)].

The Binomial Coefficient for Negative Arguments

The definition of the binomial coefficient in terms of gamma functions also allows non-integer arguments. For nonnegative integer arguments the gamma functions reduce to factorials, leading to the well-known Pascal triangle. Using a symmetry formula for the gamma function, this definition is extended to negative integer arguments, making the symmetry identity for binomial coefficients valid for all integer arguments. The agreement of this definition with some other identities and with the binomial theorem is investigated.

$q$-有理与$q$-实二项式系数

$q$-Rational and $q$-Real Binomial Coefficients

本文研究Morier-Genoud和Ovsienko用连分数定义的$q$-有/实数构造的$q$-二项式系数，建立对应$q$-帕斯卡恒等式与$q$-二项式定理，并推导更多相关恒等式。

涉及二项式系数与特殊数的双重求和

Double sums involving binomial coefficients and special numbers

该研究提出一种初等方法处理内层为不完全二项式和的双重求和问题，并将核心恒等式应用于含调和数、Fibonacci数等多种特殊数的双重求和场景。

反超几何参数扩展在递归二项式和中的形式反驳

A Formal Refutation of the Hypergeometric Parametric Extension for Reciprocal Binomial Sums

本文通过逻辑一致性检查、积分推导分析和符号计算，证明Pain提出的参数扩展在递归二项式和中的超几何闭形式表示是错误的。

Flagged Schur多项式的对偶性：经由格路径双射

Flagged Schur polynomial duality via a lattice path bijection

本文通过在推广的格上进行路径计数论证，证明了Flagged Schur多项式的恒等式，揭示了行列flag的对偶性，并推广了Gessel-Viennot与Aitken的经典对偶定理。

Some congruences involving central q-binomial coefficients

Motivated by recent works of Sun and Tauraso, we prove some variations on the Green-Krammer identity involving central q-binomial coefficients, such as $$ \sum_{k=0}^{n-1}(-1)^kq^{-{k+1\choose 2}}{2k\brack k}_q \equiv (\frac{n}{5}) q^{-\lfloor n^4/5\rfloor} \pmod{Φ_n(q)}, $$ where $\big(\frac{n}{p}\big)$ is the Legendre symbol and $Φ_n(q)$ is the $n$th cyclotomic polynomial. As consequences, we deduce that $$ \sum_{k=0}^{3^a m-1} q^{k}{2k\brack k}_q &\equiv 0 \pmod{(1-q^{3^a})/(1-q)}, \sum_{k=0}^{5^a m-1}(-1)^kq^{-{k+1\choose 2}}{2k\brack k}_q &\equiv 0 \pmod{(1-q^{5^a})/(1-q)}, $$ for $a,m\ge…

q-皮夫-萨尔舒茨恒等式的新的双射证明及其在量子群中的应用

A new bijective proof of the $q$-Pfaff--Saalschütz identity with applications to quantum groups

本文通过双射方法证明了q-皮夫-萨尔舒茨恒等式，并由此推导出量子二项系数的新乘法规则及量子群的新的整形式呈现。

Exact conditions for countable inclusion-exclusion identity and extensions

We give simple necessary and sufficient conditions for the inclusion-exclusion identity to hold for an infinite countable number of sets. In terms of a random variable, whose range are nonnegative integers, this condition is equivalent to the convergence to zero of binomial moments. Some standard extensions of the countable inclusion-exclusion identity are also given.

一个具有应用价值的新三角恒等式

A new trigonometric identity with applications

该研究提出一个新的三角恒等式，证明其与第二类Chebyshev多项式相关恒等式等价，并据此推导出涉及Bernoulli多项式的求和恒等式。

基本超几何级数的正性与 MacMahon 恒等式

Positivity of the basic hypergeometric series and the MacMahon identity

本文研究满足特定约束的基本超几何级数归一化后的正多项式，给出其三个公式：MacMahon 恒等式的推广、多重集排列统计量的生成函数及有限和表达式。

关于 Chaundy 与 Bullard 恒等式 III：基本与椭圆推广

On an identity of Chaundy and Bullard. III. Basic and elliptic extensions

本文利用加权格路径模型给出 Chaundy–Bullard 恒等式的基本与椭圆推广，并由此得到二项式定理的椭圆交换变量新推广。

Short note on the convolution of binomial coefficients

We know [Rui Duarte and António Guedes de Oliveira, New developments of an old identity, manuscript arXiv:1203.5424, submitted.] that, for every non-negative integer numbers $n,i,j$ and for every real number $\ell$, $$ \sum_{i+j=n} \binom{2i-\ell}{i} \binom{2j+\ell}{j} = \sum_{i+j=n}\binom{2i}{i} \binom{2j}{j}, $$ which is well-known to be $4^n$. We extend this result by proving that, indeed, $$ \sum_{i+j=n} \binom{ai+k-\ell}{i} \binom{aj+\ell}{j} = \sum_{i+j=n} \binom{ai+k}{i} \binom{aj}{j} $$ for every integer $a$ and for every real $k$, and present new expressions for this value.

B型对合的不动点Worpitzky恒等式与正二项变换

A Fixed-Point Worpitzky Identity and a Positive Binomial Transform for Type $B$ Involutions

本文推导B型对合的不动点细化Worpitzky恒等式，经变量替换得到正二项变换，证明固定循环型层γ正性，对应经典定理并给出低循环层显式公式。

Berkovich与Uncu一个恒等式的双射证明

A bijective proof of an identity of Berkovich and Uncu

针对Fu和Tang提出的需为Berkovich与Uncu的分拆生成函数恒等式找直接组合证明的问题，本文沿Vandervelde、Fu和Tang的思路给出了该恒等式的双射证明。

广义$n$-级数与de Rham复形

Generalized $n$-series and de Rham complexes

本文研究了交换环上广义$n$-级数的代数与组合性质，证明了经典组合恒等式的$s$-类似物，并定义了导数的$s$-类似物以证明仿射直线上的Poincaré引理与Cartier同构。