Scott Ahlgren, Shalosh B. Ekhad, Ken Ono, Doron Zeilberger
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.
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For any odd prime p we obtain q-analogues of Van Hamme's supercongruence: $$ \sum_{k=0}^{\frac{p-1}{2}}{2k\choose k}^3\frac{1}{64^k} \equiv 0 \pmod{p^2} \quad\text{for}\quad p\equiv 3\pmod 4, $$ and Rodriguez-Villegas' Beukers-like supercongruences involving products of three binomial coefficients. For example, we prove that \begin{align*} \sum_{k=0}^{\frac{p-1}{2}} {2k\brack k}_{q^2}^3 \frac{q^{2k}}{(-q^2;q^2)_k^2 (-q;q)_{2k}^2} &\equiv 0\pmod{[p]^2} \quad\text{for}\quad p\equiv 3\pmod 4, \\ \sum_{k=0}^{\frac{p-1}{2}}{2k\brack k}_{q^3}\frac{(q;q^3)_k (q^{2};q^3)_{k} q^{3k} }{ (q^{6};q^{6})_k^…
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.
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.
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.
In 2002 Zhi-Wei Sun [Integers 2(2002)] published a curious identity involving binomial coefficients. In this paper we present a generalization of the identity.
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…
Using a property of the q-shifted factorial, an identity for q-binomial coefficients is proved, which is used to derive the formulas for the q-binomial coefficient for negative arguments. The result is in agreement with an earlier paper about the normal binomial coefficient for negative arguments. Some new q-binomial summation identities are derived, and the formulas for negative arguments transform some of these summation identities into each other. One q-binomial summation identity is transformed into a new q-binomial summation identity.
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.
It is shown that the curious identity of Simons follows immediately from Euler's series transformation formula and also from an identity due to Ljunggren. The relation of Simons' identity to Legendre's polynomials is also discussed. At the end we use the generalized Euler series transformation to obtain two recent binomial identities of Munarini.
The Vandermonde-Chu Binomial Coefficients Identity is shown to imply Bombieri's deep norm inequalities, via identities of Beauzamy-Dégot, and Reznick.
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。
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.
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.
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。
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 …
We derive a new Fibonacci identity. This single identity subsumes important known identities such as those of Catalan, Ruggles, Halton and others, as well as standard general identities found in the books by Vajda, Koshy and others. We also derive several binomial and ordinary summation identities arising from this identity; in particular we obtain a generalization of Halton's general Fibonacci identity.
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。
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)].
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.
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.
$q$-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。
We prove a conjecture of Liu and Wang on the q-log-convexity of the polynomial sequence $\{\sum_{k=0}^n{n\choose k}^2q^k\}_{n\geq 0}$. By using Pieri's rule and the Jacobi-Trudi identity for Schur functions, we obtain an expansion of a sum of products of elementary symmetric functions in terms of Schur functions with nonnegative coefficients. Then the principal specialization leads to the q-log-convexity. We also prove that a technical condition of Liu and Wang holds for the squares of the binomial coefficients. Hence we deduce that the linear transformation with respect to the triangular arra…
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.
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.
In a previous paper, we studied an overpartition analogue of Gaussian polynomials as the generating function for overpartitions fitting inside an $m \times n$ rectangle. Here, we add one more parameter counting the number of overlined parts, obtaining a two-parameter generalization $\overline{m+n \brack n}_{q,t}$ of Gaussian polynomials, which is also a $(q,t)$-analogue of Delannoy numbers. First we obtain finite versions of classical $q$-series identities such as the $q$-binomial theorem and the Lebesgue identity, as well as two-variable generalizations of classical identities involving Gauss…
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…
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。
New Polynomial Identities and Some Consequences
本文借助 Euler Beta 函数与二项式定理推导两个多项式恒等式，推广已知结果，并由此得到及推广 Frisch、Klamkin 组合恒等式，还给出一类恒等式的推导方案。
In previous work on Clebsch-Gordan coefficients, certain remarkable hexagonal arrays of integers are constructed that display behaviors found in Pascal's Triangle. We explain these behaviors further using the binomial transform and discrete convolution. Here we begin by introducing the notion of a binomial array and develop several "hockey stick" rules. Then we give an algorithm that generalizes the classical Vandermonde Identity; this produces infinite families of summation formulas, which we use to expand and prove certain combinatorial identities for the Catalan numbers. Finally, we recast …
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。
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。
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.
Worpitzky's identity expresses $n^p$ in terms of the Eulerian numbers and binomial coefficients: $$n^p = \sum_{i=0}^{p-1} \genfrac<>{0pt}{}{p}{i} \binom{n+i}{p}.$$ Pita-Ruiz recently defined numbers $A_{a,b,r}(p,i)$ implicitly to satisfy a generalized Worpitzky identity $$\binom{an+b}{r}^p = \sum_{i=0}^{rp} A_{a,b,r}(p,i) \binom{n+rp-i}{rp},$$ and asked whether there is a combinatorial interpretation of the numbers $A_{a,b,r}(p,i)$. We provide such a combinatorial interpretation by defining a notion of descents in colored multipermutations, and then proving that $A_{a,b,r}(p,i)$ is equal to th…
We use the method of tiling to give elementary combinatorial proofs of some celebrated $q$-series identities, such as Jacobi triple product identity, Rogers-Ramanujan identities, and some identities of Rogers. We give a tiling proof of the q-binomial theorem and a tiling interpretation of the q-binomial coefficients. A new generalized $ k $-product $q$-series identity is also obtained by employing the `tiling-method', wherein the generating function of the set of all possible tilings of a rectangular board is computed in two different ways to obtain the desired $q$-series identity. Several new…
Shifted-Binomial Expansions of Dilated Binomial Polynomials: Multinomial Decimation, Reflection Symmetry, and Universal Divisibility
研究多项式\\(g_{p,r}(n)=\binom{pn}{p+r}\\)在移位二项式基下的展开，通过生成函数等得到系数公式，刻画其回文性及相关性质，证明\\(p\\)整除系数，还发现与多项式三角形选定行一致，揭示了二项式系数展开的一些规律。
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。
Proof of Two Multivariate $q$-Binomial Sums Arising in Gromov-Witten Theory
本研究证明了Gromov-Witten理论中两个与特定对数Calabi-Yau曲面不变量生成级数相关的多元$q$-二项式猜想恒等式，核心工具是基本超几何级数中的Jackson的Pfaff-Saalschütz求和公式的$q$-模拟。
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)].
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…
A New Proof of Fine's Identity using Wildberger's Polynomial Formula
本文通过将Wildberger和Rubine的subdigons推广为tubdigons，并对其给定类型进行双重计数，比较对应项后为Fine恒等式提供了一个新证明。
Unimodular matrices and lattice paths enumeration via Pascal's triangle
研究一类由二项式系数定义的矩阵，证明其行列式恒为1，揭示了帕斯卡三角形与格路计数之间的深刻联系。
We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler's Telescoping Lemma, we give alternate proofs of all the key summation theorems for terminating Hypergeometric Series and Basic Hypergeometric Series, including the terminating Binomial Theorem, the Chu--Vandermonde sum, the Pfaff--Saalch\" utz sum, and their $q$-analogues. We also give a proof of Jackson's $q$-analog of Dougall's sum, the sum of a terminating, balanced, very-well-poised $_8ϕ_7$ sum. …
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…
Andrews--Gordon and Stanton type identities: bijective and Bailey lemma approaches
本文利用双射方法、Bailey 引理和 Bailey 格为 Stanton 提出的 Andrews--Gordon 与 Bressoud 恒等式的两类推广给出新证明，并由此自然导出若干经典恒等式的新 Stanton 型推广。
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。
A Proof of Sundaram's Bounded-Interval Higher Lie Positivity Conjecture
证明森德拉姆关于\(F_{n,M}\)舒尔展开式系数非负整数的猜想，通过幂和展开、伯恩斯坦公式等方法，利用杨图划分、特征值分离等技术，还给出相关反例。
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