Shaoshi Chen
A Wilf--Zeilberger pair $(F, G)$ in the discrete case satisfies the equation $ F(n+1, k) - F(n, k) = G(n, k+1) - G(n, k)$. We present a structural description of all possible rational Wilf--Zeilberger pairs and their continuous and mixed analogues.
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In 1992, Wilf and Zeilberger conjectured that a hypergeometric term in several discrete and continuous variables is holonomic if and only if it is proper. Strictly speaking the conjecture does not hold, but it is true when reformulated properly: Payne proved a piecewise interpretation in 1997, and independently, Abramov and Petkovsek in 2002 proved a conjugate interpretation. Both results address the pure discrete case of the conjecture. In this paper we extend their work to hypergeometric terms in several discrete and continuous variables and prove the conjugate interpretation of the Wilf-Zei…
In this paper, we prove some supercongruences via the Wilf-Zeilberger method. For instance, for any odd prime $p$ and positive integer $r$ and $δ\in\{1,2\}$, we have \begin{align*} \sum_{n=0}^{(p^r-1)/δ} \frac{\left(\frac12\right)^5_n}{n!^5}(10n^2+6n+1)(-4)^n &\equiv\begin{cases}p^{2r}\ \pmod{p^{r+4}} &\tt{if}\ r\leq4, \\0\ \pmod{p^{r+4}} &\tt{if}\ r \geq5. \end{cases} \end{align*}
Wilf-Zeilberger seeds and non-trivial hypergeometric identities
本研究采用生成Wilf-Zeilberger对的系统方法，证明了多位学者提出的超几何恒等式猜想，涵盖Ramanujan型1/π相关公式及ζ(5)级数等重要结果。
Discovering hypergeometric series with harmonic numbers via Wilf-Zeilberger seeds
本文通过Wilf-Zeilberger种子方法发现包含调和数的超几何级数，并首次获得深度为二的多重ζ值ζ(5,3)的快速收敛级数，同时推测WZ-种子背后的格雷德代数结构。
In this paper, we prove two supercongruences by the Wilf-Zeilberger method. One of them is, for any prime $p>3$, \begin{align*} \sum_{n=0}^{(p-1)/2}\frac{3n+1}{(-8)^n}\binom{2n}n^3\equiv p\left(\frac{-1}p\right)+\frac{p^3}4\left(\frac2p\right)E_{p-3}\left(\frac14\right)\pmod{p^4}, \end{align*} where $\left(\frac{\cdot}p\right)$ stands for the Legendre symbol, and $E_{n}(x)$ are the Euler polynomials. This congruence confirms a conjecture of Sun \cite[(2.18)]{sun-numb-2019} with $n=1$.
$q$-analogues of Wilf-Zeilberger seeds and Ramanujan $1/π^k$-formulas
本文提出Wilf-Zeilberger种子的概念，用于生成$q$-类比的Ramanujan型$1/π^k$公式，并揭示了部分非模形式的模伪模实例。
One of the most challenging problems in enumerative combinatorics is to count Wilf classes, where you are given a pattern, or set of patterns, and you are asked to find a "formula", or at least an efficient algorithm, that inputs a positive integer n and outputs the number of permutations avoiding that pattern. In 1996, John Noonan and Doron Zeilberger initiated the counting of permutations that have a prescribed, r, say, occurrences of a given pattern. They gave an ingenious method to generate Functional Equations, alas, with an unbounded number of "catalytic variables", but then described a …
We prove that the power series expansion of the rational function of Gillis, Reznick and Zeilberger (1983) has only nonnegative coefficients.
By using the Wilf-Zeilberger method, we prove a novel finite combinatorial identity related to a bivariate generating function for $ζ(2+r+2s)$ (an extension of a Bailey-Borwein-Bradley Apery-like formula for even zeta values). Such identity is then applied to show several supercongruences.
The limit of a Stanley-Wilf sequence is not always rational, and layered patterns beat monotone patterns
我们展示了模式$q$的第一个已知例子，其中$\lim_{n\to \infty} \sqrt[n]{S_n(q)}$不是整数。我们找到了极限的精确值，并证明了它是无理数。然后我们将我们的结果推广到一个无穷序列的模式。最后，我们提供了进一步的概括，开始解释为什么某些模式比其他模式更容易避免。最后，我们表明，如果$q$是一个分层模式的长度$k$，那么$L(q)\geq (k-1)^2$举行。
Multiple zeta values, WZ-pairs and infinite sums computations
本研究结合Wilf-Zeilberger对方法与多重zeta值系统理论，证明了Z.W. Sun提出的大量级数恒等式，破解了诸多长期悬而未决的猜想。
We show how to extend the Karolyi-Nagy beautiful proof of the Zeilberger-Bressoud q-Dyson theorem, (first proved by Zeilberger and Bressoud in 1985, and originally conjectured by George Andrews in 1975), that states that the constant term of a certain Laurent polynomial equals the q-multinomial coefficient, how to evaluate any other specific coefficient. The algorithm implies that any such coefficient is always a certain rational function (that the algorithm finds) times the q-multinomial coefficient.
A construction method for WZ seeds
本文提出一种系统方法构造WZ种子，并给出了七个WZ种子，展示了如何从现有种子构造WZ种子，并推导出多个超几何恒等式，该方法可扩展到q情况，得到七种q-类WZ种子。
In this paper, we prove two supercongruences conjectured by Z.-W. Sun via the Wilf-Zeilberger method. One of them is, for any prime $p>3$, \begin{align*} \sum_{n=0}^{p-1}\frac{6n+1}{256^n}\binom{2n}n^3&\equiv p(-1)^{(p-1)/2}-p^3E_{p-3}\pmod{p^4}. \end{align*} In fact, this supercongruence is a generalization of a supercongruence of van Hamme.
Using Wilf-Zeilberger algorithmic proof theory, we continue pioneering work of Meni Rosenfeld (followed up by interesting work by Cyril Grunspan and Ricardo Perez-Marco) and study the probability and duration of successful bitcoin attacks, but using an equivalent, and much more congenial, formulation as a certain two-phase soccer match.
Applicability of the $q$-Analogue of Zeilberger's Algorithm
蔡尔伯格算法的一般情况的适用性或终止条件最近由阿布拉莫夫得到。对于$q$ -类似物，二元$q$ -超几何项是否具有$qZ$ -对的问题仍然是开放的.乐已经找到了一个解决这个问题时，给定的二元$q$ -超几何项是一个有理函数在某些权力的$q$。我们解决了这个问题的一般情况下，通过给二元$q$ -超几何项的$q$ -模拟Zeilberger的算法终止的表征。此外，我们给出了一个算法，以确定一个二元$q$ -超几何项是否有一个$qZ$ -对。
In this paper, we solve the existence problem of telescopers for rational functions in three discrete variables. We reduce the problem to that of deciding the summability of bivariate rational functions, which has been solved recently. The existence criteria we present is needed for detecting the termination of Zeilberger's algorithm to the function classes studied in this paper.
Using the $q$-Wilf--Zeilberger method and a $q$-analogue of a "divergent" Ramanujan-type supercongruence, we give several $q$-supercongruences modulo the fourth power of a cyclotomic polynomial. One of them is a $q$-analogue of a supercongruence recently proved by Wang: for any prime $p>3$, $$ \sum_{k=0}^{p-1} (3k-1)\frac{(\frac{1}{2})_k (-\frac{1}{2})_k^2 }{k!^3}4^k\equiv p-2p^3 \pmod{p^4}, $$ where $(a)_k=a(a+1)\cdots (a+k-1)$ is the Pochhammer symbol.
Meeting a Challenge raised by Ekhad and Zeilberger related to Stern's Triangle
针对与 Stern 三角形相关的 ω(10000) 计算难题，论文整合常数项、条件转移矩阵、代数生成函数与 P-递归，算出该 6591 位数值。
Zeilberger's method of creative telescoping is crucial for the computer-generated proofs of combinatorial and special-function identities. Telescopers are linear differential or ($q$-)recurrence operators computed by algorithms for creative telescoping. For a given class of inputs, when telescopers exist and how to construct telescopers efficiently if they exist are two fundamental problems related to creative telescoping. In this paper, we solve the existence problem of telescopers for rational functions in three variables including 18 cases. We reduce the existence problem from the trivariat…
We give necessary and sufficient conditions for the existence of telescopers for rational functions of two variables in the continuous, discrete and q-discrete settings and characterize which operators can occur as telescopers. Using this latter characterization, we reprove results of Furstenberg and Zeilberger concerning diagonals of power series representing rational functions. The key concept behind these considerations is a generalization of the notion of residue in the continuous case to an analogous concept in the discrete and q-discrete cases.
A Generalized Supercongruence of Z.-W. Sun
本文利用Wilf-Zeilberger方法证明了Z.-W. Sun提出的超同余猜想，针对任意素数p，通过组合恒等式和符号求和技术，验证了特定求和表达式模p³的同余关系。
S. Ekhad and D. Zeilberger recently proved that the multivariate generating function for the number of simple singular vector tuples of a generic $m_1 \times \cdots \times m_d$ tensor has an elegant rational form involving elementary symmetric functions, and provided a partial conjecture for the asymptotic behavior of the cubical case $m_1 = \cdots = m_d$. We prove this conjecture and further identify completely the dominant asymptotic term, including the multiplicative constant. Finally, we use the method of differential approximants to conjecture that the subdominant connective constant effe…
We consider asymptotics of power series coefficients of rational functions of the form $1/Q$ where $Q$ is a symmetric multilinear polynomial. We review a number of such cases from the literature, chiefly concerned either with positivity of coefficients or diagonal asymptotics. We then analyze coefficient asymptotics using ACSV (Analytic Combinatorics in Several Variables) methods. While ACSV sometimes requires considerable overhead and geometric computation, in the case of symmetric multilinear rational functions there are some reductions that streamline the analysis. Our results include diago…
Experimenting with Discrete Dynamical Systems
本文利用实验数学和符号计算，严格或半严格地证明了关于有理差分方程的若干猜想，并扩展了证明任意正系数有理差分方程全局渐近稳定性的方法。
We investigate a generalization of stacks that we call $\mathcal{C}$-machines. We show how this viewpoint rapidly leads to functional equations for the classes of permutations that $\mathcal{C}$-machines generate, and how these systems of functional equations can frequently be solved by either the kernel method or, much more easily, by guessing and checking. General results about the rationality, algebraicity, and the existence of Wilfian formulas for some classes generated by $\mathcal{C}$-machines are given. We also draw attention to some relatively small permutation classes which, although …
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