利用Zeilberger算法推广Swisher的(G.3)型一个猜想超同余式

On two conjectural supercongruences of Apagodu and Zeilberger

Let the numbers $α_n,β_n$ and $γ_n$ denote \begin{align*} α_n=\sum_{k=0}^{n-1}{2k\choose k},\quad β_n=\sum_{k=0}^{n-1}{2k\choose k}\frac{1}{k+1}\quad\text{and}\quad γ_n=\sum_{k=0}^{n-1}{2k\choose k}\frac{3k+2}{k+1}, \end{align*} respectively. We prove that for any prime $p\ge 5$ and positive integer $n$ \begin{align*} α_{np}&\equiv \left(\frac{p}{3}\right) α_n \pmod{p^2},\\ β_{np}&\equiv \begin{cases} \displaystyle β_n \pmod{p^2},\quad &\text{if $p\equiv 1\pmod{3}$},\\ -γ_n \pmod{p^2},\quad &\text{if $p\equiv 2\pmod{3}$}, \end{cases} \end{align*} where $\left(\frac{\cdot}{p}\right)$ denotes th…

进一步的(C.2)和(G.2)超同余的q-推广

A further $q$-generalization of the (C.2) and (G.2) supercongruences of Van Hamme

本文利用q-Zeilberger算法，建立(C.2)和(G.2)超同余的q-推广，推导出涉及伯努利数的q-超同余，并改进(G.2)超同余结果。

A bivariate generating function for zeta values and related supercongruences

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.

Supercongruences concerning truncated hypergeometric series

Let $n\geq 3$ be an integer and $p$ be a prime with $p\equiv 1\pmod{n}$. In this paper, we show that $${}_nF_{n-1}\bigg[\begin{matrix} \frac{n-1}{n}&\frac{n-1}{n}&\ldots&\frac{n-1}{n}\\ &1&\ldots&1\end{matrix}\bigg | \, 1\bigg]_{p-1}\equiv -Γ_p\bigg(\frac{1}{n}\bigg)^n\pmod{p^3}, $$ where the truncated hypergeometric series $$_nF_{n-1} \bigg[\begin{matrix} x_1&x_2&\ldots&x_n\\ &y_1&\cdots&y_{n-1}\end{matrix}\bigg | \, z\bigg]_m=\sum_{k=0}^{m}\frac{z^k}{k!}\prod_{j=0}^{k-1}\frac{(x_1+j)\cdots(x_{n}+j)}{(y_1+j)\cdots(y_{n-1}+j)} $$ and $Γ_p$ denotes the $p$-adic gamma function. This confirms a c…

Proof of two supercongruences conjectured by Z.-W. Sun

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.

Some variations of a "divergent" Ramanujan-type $q$-supercongruence

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.

关于涉及对偶序列 $s_n(x)$ 的一个猜想性超同余

On a conjectural supercongruence involving the dual sequence $s_n(x)$

本文证实了Z.-W. Sun提出的关于对偶序列$s_n(x)$平方和模$p^3$的一个猜想性超同余，该猜想源于Kimoto-Wakayama超同余的推广。

截断超几何级数 ${}_4F_3$ 的两个超同余式的证明

Proof of two supercongruences of truncated hypergeometric series ${}_4F_3$

本文利用 Wilf-Zeilberger 方法证明了孙智伟提出的两个关于截断超几何级数 ${}_4F_3$ 的超同余式，并给出其中一个涉及勒让德符号与欧拉数的模 $p^4$ 同余结果。

通过 $q$-Lucas 定理的 Dwork 型 $q$-同余

Dwork-type $q$-congruences through the $q$-Lucas theorem

本文运用 $q$-Lucas 定理和已知 $q$-超同余，证明了若干 Dwork 型 $q$-同余，证实了三个先前提出的猜想，并验证了 Swisher (H.3) 猜想的较弱形式。

Some congruences related to a congruence of Van Hamme

We establish some supercongruences related to a supercongruence of Van Hamme, such as \begin{align*} \sum_{k=0}^{(p+1)/2} (-1)^k (4k-1)\frac{(-\frac{1}{2})_k^3}{k!^3} &\equiv p(-1)^{(p+1)/2}+p^3(2-E_{p-3})\pmod{p^{4}},\\ \sum_{k=0}^{(p+1)/2} (4k-1)^5 \frac{(-\frac{1}{2})_k^4}{k!^4} &\equiv 16p\pmod{p^{4}}, \end{align*} where $p$ is an odd prime and $E_{p-3}$ is the $(p-3)$-th Euler number. Our proof uses some congruences of Z.-W. Sun, the Wilf--Zeilberger method, Whipple's $_7F_6$ transformation, and the software package {\tt Sigma} developed by Schneider. We also put forward two related conje…