David Callan
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.
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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.
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.
In this note, we present a curious $q$-series identity with applications to certain partitions with bounded part differences.
We show a curious identity on root systems which gives the evaluation of the volume of the spherical simpleces cut by the cone generated by simple roots.
We prove that a curious generating series identity implies Faber's intersection number conjecture (by showing that it implies a combinatorial identity already given in arXiv:1902.02742) and give a new proof of Faber's conjecture by directly proving this identity.
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。
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.
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.
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.
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。
In this paper, we prove a conjecture raised by Zhi-Wei Sun in 2018 by the eigenvectors-eigenvalues identity found by Denton, Parke, Tao and X. Zhang in 2019.
We evaluate a curious determinant, first mentioned by George Andrews in 1980 in the context of descending plane partitions. Our strategy is to combine the famous Desnanot-Jacobi-Dodgson identity with automated proof techniques. More precisely, we follow the holonomic ansatz that was proposed by Doron Zeilberger in 2007. We derive a compact and nice formula for Andrews's determinant, and use it to solve a challenge problem that we posed in a previous paper. By noting that Andrews's determinant is a special case of a two-parameter family of determinants, we find closed forms for several one-para…
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.
In this paper, we prove several supercongruences conjectured by Z.-W. Sun ten years ago via certain strange hypergeometric identities. For example, for any prime $p>3$, we show that $$\sum_{k=0}^{p-1}\frac{\binom{4k}{2k+1}\binom{2k}k}{48^k}\equiv0\pmod{p^2},$$ and $$ \sum_{k=0}^{p-1}\frac{\binom{2k}{k}\binom{3k}{k}}{24^k}\equiv\begin{cases}\binom{(2p-2)/3}{(p-1)/3}\pmod{p^2}\ &\mbox{if}\ p\equiv1\pmod{3},\\ p/\binom{(2p+2)/3}{(p+1)/3}\pmod{p^2}\ &\mbox{if}\ p\equiv2\pmod{3}.\end{cases} $$ We also obtain some other results of such types.
Recently, Guo and Sun derived an identity for factorial Grothendieck polynomials which is a generalization of the one for Schur polynomials by Fehér, Némethi and Rimányi. We analyze the identity from the point of view of quantum integrability, based on the correspondence between the wavefunctions of a five-vertex model and the Grothendieck polynomials. We give another proof using the quantum inverse scattering method. We also apply the same idea and technique to derive an identity for factorial Grothendieck polynomials for rectangular Young diagrams. Combining with the Guo-Sun identity, we get…
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…
We give bijective proofs of orthosymplectic analogues of the Cauchy identity and dual Cauchy identity for orthosymplectic Schur functions. To do so, we present two insertion algorithms; these are orthosymplectic versions of Berele's symplectic insertion algorithms, which were used by Sundaram to give bijective proofs of Cauchy identities for symplectic Schur functions.
We give a new proof of a theorem of Mansour and Sun by using number theory and Rothe's identity.
Motivated by the recent work of Chu [Electron. J. Combin. 17 (2010), #N24], we give simple proofs of Jensen's identity $$ \sum_{k=0}^{n}{x+kz\choose k}{y-kz\choose n-k} =\sum_{k=0}^{n}{x+y-k\choose n-k}z^k, $$ and Chu's and Mohanty-Handa's generalizations of Jensen's identity. We also give a quite simple proof of an equivalent form of Graham-Knuth-Patashnik's identity $$ \sum_{k\geq 0}{m+r\choose m-n-k}{n+k\choose n}x^{m-n-k}y^k =\sum_{k\geq 0}{-r\choose m-n-k}{n+k\choose n}(-x)^{m-n-k}(x+y)^k, $$ which was rediscovered, respectively, by Sun in 2003 and Munarini in 2005. Finally we give a mult…
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。
An elementary approach to Sun Kim's general theta function identities
本文提出初等方法，在统一框架下用 q-级数证明 Kim 的一般 theta 函数恒等式，这些恒等式涵盖度数 5、11、23 的模方程及已知划分定理。
A further $q$-analogue of a formula due to Guillera
本文构造并证明了Guillera关于π²公式的一个新的q-模拟，该模拟与已有结果不等价，且额外引入一个自由参数，推导基于Zeilberger算法的q-模拟。
According to the $q$-series method, a short proof for Hou and Sun's identity, which is the $q$-analogue of a known $π$-formula, is offered. Furthermore, $q$-analogues of several other $π$-formulas are also established in terms of the $q$-series method and the telescoping method.
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.
The initial purpose of this paper is to provide a combinatorial proof of the minor summation formula of Pfaffians based on the lattice path method. There we related Plücker relations with the minor summation formula of Pfaffians to simplify its proof by lattice paths. The second aim is to find a various applications of the minor summation formula. First we studied a variant of the Sundquist formula established in J. Alg. Combin. {\bf 5} (1996). Next we gave a simple proof of Kawanaka's formula concerning a $q$-series identity involving Schur functions in Osaka J. Math. {\bf 36} (1999). We also…
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…
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