Congruences related to Miki's identity

A new series for $π^3$ and related congruences

Let $H_n^{(2)}$ denote the second-order harmonic number $\sum_{0<k\le n}1/k^2$ for $n=0,1,2,\ldots$. In this paper we obtain the following identity: $$\sum_{k=1}^\infty\frac{2^kH_{k-1}^{(2)}}{k\binom{2k}k}=\frac{π^3}{48}.$$ We explain how we found the series and develop related congruences involving Bernoulli or Euler numbers; for example, it is shown that $$\sum_{k=1}^{p-1}\frac{\binom{2k}k}{2^k}H_k^{(2)}\equiv-E_{p-3}\pmod{p}$$ for any prime $p>3$, where $E_0,E_1,E_2,\ldots$ are Euler numbers. Motivated by the Amdeberhan-Zeilberger identity $\sum_{k=1}^\infty(21k-8)/(k^3\binom{2k}k^3)=π^2/6$…

Banach matchboxes problem and a congruence for primes

Using an identity arising in the known Banach probability problem on matchboxes, we prove an unexpected congruence for odd prime $p:$ for $1\leq k\leq \frac{p-1}{2},\enskip \sum_{i=1}^{p-2k-1}2^{i-1}\binom{k-1+i}{k}\equiv 0\pmod p.$

A binomial coefficient identity associated to a conjecture of Beukers

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.

广义中心三项式系数的参数化 Catalan 型同余

A parametric Catalan-type congruence for generalized central trinomial coefficients

本文针对广义中心三项式系数，给出一个参数化 Catalan 型同余式，解决 Wang 和 Cui 的求和问题，并覆盖 $p$ 整除 $b$ 或 $b-2c$ 的情形。

按奇偶性分隔部件的分拆：共轭、同余式与 mock theta 函数

Partitions with parts separated by parity: conjugation, congruences and the mock theta functions

本文利用共轭映射的组合方法研究奇偶部件分隔的整数分拆，证明推广 Andrews 二元生成函数的参数化恒等式与两族 Andrews–Beck 型同余式，并揭示共轭稳定分拆子集同三个三阶 mock theta 函数的联系。

对Andrews和Newman关于最小排除项之和结果的一个精细化

A refinement of a result of Andrews and Newman on the sum of minimal excludants

本文精细化Andrews和Newman关于最小排除项之和的恒等式，得到相关函数的模4和模8同余式，并提出新的同余猜想，同时开始研究最小排除项的高阶矩。

奇部超额分拆元组的模$2$幂内部同余

Internal congruences modulo powers of $2$ for overpartition tuples with odd parts

本文证明奇部超额分拆元组数在$2$幂指标下的内部同余，通过统一的三项递推多项式族给出初等证明，并精确刻画$2$-进赋值等于奇平方数。

The Rogers-Ramanujan-Gordon Theorem for Overpartitions

Let $B_{k,i}(n)$ be the number of partitions of $n$ with certain difference condition and let $A_{k,i}(n)$ be the number of partitions of $n$ with certain congruence condition. The Rogers-Ramanujan-Gordon theorem states that $B_{k,i}(n)=A_{k,i}(n)$. Lovejoy obtained an overpartition analogue of the Rogers-Ramanujan-Gordon theorem for the cases $i=1$ and $i=k$. We find an overpartition analogue of the Rogers-Ramanujan-Gordon theorem in the general case. Let $D_{k,i}(n)$ be the number of overpartitions of $n$ satisfying certain difference condition and $C_{k,i}(n)$ be the number of overpartition…

缺失的Moore图中无对合

No involutions in the missing Moore graph

本文证明若存在度为57的Moore图，则其无对合自同构，结合已知不动点结构与模论障碍得出矛盾。