An extension of a supercongruence of Long and Ramakrishna

Bivariate identities for values of the Hurwitz zeta function and supercongruences

In this paper, we prove a new identity for values of the Hurwitz zeta function which contains as particular cases Koecher's identity for odd zeta values, the Bailey-Borwein-Bradley identity for even zeta values and many other interesting formulas related to values of the Hurwitz zeta function. We also get an extension of the bivariate identity of Cohen to values of the Hurwitz zeta function. The main tool we use here is a construction of new Markov-WZ pairs. As application of our results, we prove several conjectures on supercongruences proposed by J. Guillera, W. Zudilin, and Z.-W. Sun.

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

Extension of a conjectural supercongruence of (G.3) of Swisher using Zeilberger's algorithm

该研究利用Zeilberger算法证明了一类对任意奇数r>3成立的超同余式，将Swisher的(G.3)第三个猜想超同余式推广到了更高阶的素数幂模。

与Van Hamme的(E.2)和(F.2)超同余相关的一些超同余的q-模拟

$q$-Analogues of some supercongruences related to Van Hamme's (E.2) and (F.2) supercongruences

本文给出Van Hamme的(E.2)和(F.2)超同余相关超同余的单参数扩展，利用creative microscoping方法和Jackson的${}_{6}\phi_{5}$求和公式推导了分圆多项式三次方模的q-超同余。

On the supercongruence conjectures of van Hamme

In 1997, van Hamme developed $p-$adic analogs, for primes $p$, of several series which relate hypergeometric series to values of the gamma function, originally studied by Ramanujan. These analogs relate truncated sums of hypergeometric series to values of the $p-$adic gamma function, and are called Ramanujan type supercongruences. In all, van Hamme conjectured 13 such formulas, three of which were proved by van Hamme himself, and five others have been proved recently using a wide range of methods. Here, we explore four of the remaining five van Hamme supercongruences, revisit some of the prove…

Gessel-Lucas congruences for sporadic sequences

For each of the $15$ known sporadic Apéry-like sequences, we prove congruences modulo $p^2$ that are natural extensions of the Lucas congruences modulo $p$. This extends a result of Gessel for the numbers used by Apéry in his proof of the irrationality of $ζ(3)$. Moreover, we show that each of these sequences satisfies two-term supercongruences modulo $p^{2r}$. Using special constant term representations recently discovered by Gorodetsky, we prove these supercongruences in the two cases that remained previously open.

p-adic刚性及秩二吸引子处的超同余

p-adic Rigidity and Supercongruences at Rank Two Attractors

本文研究秩二吸引子引发的p-adic刚性，提出上同调判据证明Frobenius不动点与超同余，并应用于Hulek--Verrill A_4族及镜像四次K3铅笔。

具有给定 Galois 群的最大非分歧 $p$-扩张：Ozaki 定理的定量改进

Maximal Unramified $p$-Extensions with Prescribed Galois Groups: A Quantitative Refinement of Ozaki's Theorem

本文改进了 Ozaki 定理，通过利用 Frattini 结构将实现有限 p-群为 p-类塔群所需的基域次数从阶尺度降至二次尺度，并给出初等交换群的精确估计。