Identities and congruences for a new sequence
Zhi-Hong Sun
令$[x]$为不超过$x$的最大整数。本文引入了由$U_0=1$和$U_n=-2\sum_{k=1}^{[n/2]}\binom n{2k}U_{n-2k}\quad(n\ge 1)$给出的序列$\{U_n\}$，并建立了涉及$\{U_n\}$的许多递归公式和同余式.
Let $[x]$ be the greatest integer not exceeding $x$. In the paper we introduce the sequence $\{U_n\}$ given by $U_0=1$ and $U_n=-2\sum_{k=1}^{[n/2]}\binom n{2k}U_{n-2k}\quad(n\ge 1)$, and establish many recursive formulas and congruences involving $\{U_n\}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
For the unrestricted partition function $p(n)$ for integers $n \geq 0$, it is known that $p(49n + 19) \equiv 0 \pmod{49}$, $p(49n + 33) \equiv 0 \pmod{49}$, and $p(49n + 40) \equiv 0 \pmod{49}$ for all $n \geq 0$. We find witness identities for these Ramanujan congruences.
Congruence identities of regularized multiple zeta values involving a pair of index sets
利用嵌套和将Riemann zeta值推广为多重zeta值（MZVs），通过发散无穷级数的正则化将多重zeta值推广为正则化多重zeta值（RMZVs）.本文证明了深度为$n-1$的RMZV与其乘积所张成的向量空间中包含一对指标集的深度为$n$的RMZV的同余同余关系.我们还得到了奇偶性结果的一个证明，以及MZVs的一个同余和公式。
Identities and congruences involving the Fubini polynomials
本文研究了Fubini多项式$F_{x}^{n}:=F_{n}(x)$的阴影表示，得到了涉及这类多项式的一些性质.对于任意素数$p$和任意整系数多项式$f$，我们证明了$(f(F_{x}))^{p}\equiv f(F_{x})$，并给出了其它有趣的同余式.
Some New Congruences and Partition-Theoretic Interpretations for the Coefficients of Some Rogers-Ramanujan Type Identities
本文利用正整数的 overpartition 和 colour partition，为某些 Rogers-Ramanujan 型恒等式系数提供了分拆理论解释，并证明了模 2 的幂的无限族同余。
In this paper, near-miss identities for the number of representations of some integral ternary quadratic forms with congruence conditions are found and proven. The genus and spinor genus of the corresponding lattice cosets are then classified. Finally, a complete genus and spinor genus classification for all conductor 2 lattice cosets of 2-adically unimodular lattices is given.
Congruence Identities Arising From Dynamical Systems
通过对某些区间映射的所有周期的周期点的个数进行计数，得到了无穷多个新的数论同余恒等式。
Polynomial identities and Fermat quotients
本文通过证明多项式恒等式，推导出费马商模 $p^2$ 的更简单同余式，推广 Eisenstein 经典同余式，并得到调和数相关和式及高阶二项和公式。
A Simple Method Which Generates Infinitely Many Congruence Identities
本文介绍了一种简单的方法--实直线上分段线性函数的符号表示法，并用它计算了某些分段线性函数的所有周期的周期点的个数。由于对每一个正整数m，最小周期m的周期点的个数必能被m整除，我们得到了无穷多个同余恒等式。
Identities and congruences involving orthogonal polynomials and Apéry-like numbers
本文建立三项递推序列的一般恒等式，应用于正交多项式和Apéry类数，给出相关恒等式与同余式。
Arithmetic Identities and Congruences for Partition Triples with 3-cores
令${{B}_{3}}(n)$表示$n$的分区三元组的数量，其中每个分区是3核的。借助于生成函数的操作，我们找到了${{B}_{3}}(n)$的算术恒等式和同余式的几个无穷族。另外，设$ω(n)$表示非负整数$n$的表示数，形式为$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+3y_{1}^{2}+3y_{2}^{2}+3y_{3}^{2}$ with ${{x}_{1}},{{x}_{2}},{{x}_{3}},{{y}_{1}},{{y}_{2}},{{y}_{3}}\in \mathbb{Z}.$，我们发现了${{B}_{3}}(n)$与$ω(n)$之间的三种算术关系，如$ω(6n+5)=4{{B}_{3}}(6n+4).$
Some new congruences and identities for $SOME(n)$, $DSOME(n)$, $\overline{SOME}(n)$ functions and analogues
本文证明了 $SOME(n)$, $DSOME(n)$, $\overline{SOME}(n)$ 函数的新恒等式和同余式（包括单调性），并推广到一般分拆族 $S_{\mathcal P}(n)$ 和彩色分拆类似物，证明了可除性性质。
Two partition functions with congruences modulo 3, 5, 7, and 13
我们引入了两个新的整数分拆函数，这两个函数都是具有一定大小限制的$n$的分割四元组的数目。利用广义Lambert级数恒等式和$q$ -级数技巧，证明了这两个函数满足模$3$，$5$，$7$和$13$的Ramanujan型同余.
Let $B_{l,m}(n)$ denote the number of $(l,m)$-regular bipartitions of $n$. Recently, many authors proved several infinite families of congruences modulo $3$, $5$ and $11$ for $B_{l,m}(n)$. In this paper, using theta function identities to prove infinite families of congruences modulo $m$ for $(l,m)$-regular bipartitions, where $m\in\{7,3,11,13,17\}$.
In this paper, by constructing some identities, we prove some $q$-analogues of some congruences. For example, for any odd integer $n>1$, we show that \begin{gather*} \sum_{k=0}^{n-1} \frac{(q^{-1};q^2)_k}{(q;q)_k} q^k \equiv (-1)^{(n+1)/2} q^{(n^2-1)/4} - (1+q)[n] \pmod{Φ_n(q)^2},\\ \sum_{k=0}^{n-1}\frac{(q^3;q^2)_k}{(q;q)_k} q^k \equiv (-1)^{(n+1)/2} q^{(n^2-9)/4} + \frac{1+q}{q^2}[n]\pmod{Φ_n(q)^2}, \end{gather*} where the $q$-Pochhanmmer symbol is defined by $(x;q)_0=1$ and $(x;q)_k = (1-x)(1-xq)\cdots(1-xq^{k-1})$ for $k\geq1$, the $q$-integer is defined by $[n]=1+q+\cdots+q^{n-1}$ and $Φ_…
For any non-negative integer $n$ and non-zero integer $r$, let $p_r(n)$ denote Ramanujan's general partition function. By employing $q$-identities, we prove some new Ramanujan-type congruences modulo 5 for $p_r(n)$ for $r=-(5λ+1), -(5λ+3), -(5λ+4), -(25λ+1)$, $-(25λ+2)$, and any integer $λ$.
A partition of $n$ is $l$-regular if none of its parts is divisible by $l$. Let $b_l(n)$ denote the number of $l$-regular partitions of $n$. In this paper, using theta function identities due to Ramanujan, we establish some new infinite families of congruences for $b_l(n)$ modulo $l$, where $l=13,17,23$.
Some Congruences of a Restricted Bipartition Function
令$c_N(n)$表示正整数$n$的二分划$(λ, μ)$的数目，该正整数受到$μ$的每一部分可被$N$整除的限制。本文利用Ramanujan的θ-函数恒等式证明了函数$c_N(n)$对$N=7$，11和$5l$对任意整数$l\ge 1$的一些同余性质.
The congruence of Wolstenholme and generalized binomial coefficients related to Lucas sequences
利用关于基本Lucas序列的广义二项式系数，我们建立了推广经典的Wolstenholme同余和其他相关的强同余的同余.
Polynomial reduction, designed first for hypergeometric terms, can be used to automatically prove and generate new hypergeometric identities from old ones. In this paper, we extend the reduction method to holonomic sequences. As applications, we describe an algorithmic way to prove and generate new multi-summation identities. Especially we present new families of $π$-series involving Domb numbers and Franel numbers.
In the course of the proof of the irrationality of zeta(2) R. Apery introduced numbers b_n = \sum_{k=0}^n {n \choose k}^2{n+k \choose k}. Stienstra and Beukers showed that for the prime p > 3 Apery numbers satisfy congruence b((p-1)/2) = 4a^2-2p mod p, if p = a^2+b^2 (where a is odd). Later, Zagier found some generalizations of Apery numbers, so called sporadic sequences, and recently Osburn and Straub proved similar congruences for all but one of the six Zagier's sporadic sequences (three cases were already known to be true) and conjectured the congruence for the sixth sequence. In this paper…
By using Andrews's explicit formulae of the $q$-Fibonacci sequence introduced by Schur, we prove certain congruences of the $q$-Fibonacci sequence which relate the sequence with the original Fibonacci sequence. As a corollary, we show that it yields a transcendental element in the $\mathbb{Q}$-algebra $\mathscr{A}$ of integers modulo infinitely large primes under the generalized Riemann hypothesis.
Hecke Relations for Eta Multipliers and Congruences of Higher-Order Smallest Parts Functions
本研究通过作用于Eisenstein-eta商族的Hecke算子推导恒等式，得到系数素数幂模同余式，应用于解决Garvan提出的高阶最小分部函数同余问题，还建立曲柄矩、秩矩与分拆函数的素数幂模联系，强化推广了已有结果。
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.
Cyclotomic Congruences and Lucas Sequences
本文研究了卢卡斯序列中素数进入点行为的限制，并推测存在强切比雪夫偏倚。
Power multiples in binary recurrence sequences: an approach by congruences
本文介绍了一种基于初等同余的方法来寻找任意二进制递归序列（q>2）中的q次幂倍数。该程序允许证明在许多情况下不存在这样的倍数。
Binomial coefficient-harmonic sum identities associated to supercongruences
利用部分分式分解方法建立了两个二项式系数--广义调和和恒等式.这些恒等式是证明许多超同余的关键要素。特别是，在其他作品的作者，他们被用来建立模$p^k$（$k>1$）之间的同余截断广义超几何级数，并在有限域上的函数扩展格林的超几何函数的$p$ -adic设置。一个专门化的这些同余之一是用来证明一个杰出的猜想罗德里格斯Villegas有关截断广义超几何级数的$p$第傅里叶系数的一个特定的模块形式。
In this paper, we consider a generalization of Horadam sequence {w_n} which is defined by the recurrence w_n = aw_n-1 + cw_n-2; if n is even, w_n = bw_n-1 + cw_n-2; if n is odd with arbitrary initial conditions w_0, w_1 and nonzero real numbers a, b, and c. We investigate some congruence properties of the generalized Horadam sequence {w_n}.
An Extension of a Congruence by Kohnen
令$p>3$是素数，令$q_p(2)=(2^{p-1}-1)/p$是$p$以2为底的费马商。最近，Z. H. Sun证明了\sum_{k=1}^{p-1} \frac{1}{k\cdot 2^k} \equiv q_p（2）- \frac{p}{2} q_p（2）^2 \pmod{p^2}是W.科南本文利用几个组合恒等式和涉及Fermat商$q_p(2)$、调和和或交错调和和的同余式，给出了上述同余式的初等证明.
Counting solutions of quadratic congruences in several variables revisited
令$N_k(n,r,\boldsymbol{a})$表示二次同余方程式$a_1x_1^2+\ldots+a_kx_k^2\equiv n$（mod $r$）的非全等解的个数，其中$\boldsymbol{a}=(a_1,\ldots,a_k)\in {\Bbb Z}^k$，$n\in {\Bbb Z}$，$r\in {\Bbb N}$ .我们给简短的直接证明某些鲜为人知的紧公式$N_k(n,r,\boldsymbol{a})$，有效的$r$奇，这可以追溯到闵可夫斯基，巴赫曼和科恩的工作。我们还推导了一些文献中没有的相关恒等式和渐近公式。
We show that the framing of $2$-sequences whose generating functions are rational integrate to $3$-sequences. To do so, we give a generalization of Wolstenholme's Theorem.
Congruences of multiple sums involving invariant sequences under binomial transform
我们将证明几个同余模幂的素数，如$$ \sum_{0<k_1<...<k_{n}<p}\leg{p-k_{n}}{3} {(-1)^{k_{n}}\over k_1... k_{n}}\equiv {lll} -{2^{n+1}+2\over 6^{n+1}} p B_{p-n-1}({1\over 3}) &\pmod{p^2} &{if $n$ is odd} -{2^{n+1}+4\over n6^n} B_{p-n}({1\over 3}) &\pmod{p} &{if $n$ is even}. $$其中$n$是一个正整数和$p$是素数，使$p>\max(n+1,3)$。
Bell Transforms of Arithmetic Functions: Euler Products, Congruences, and Polynomial Sequences
本文提出一个统一的代数框架，利用形式贝尔变换将算术函数的狄利克雷卷积与无限欧拉型乘积的组合结构联系起来，通过分析指数生成函数的对数导数，建立了贝尔指数与莫比乌斯逆运算之间的显式映射，并应用于推导经典序列的精确消失性质和同余继承性质，包括拉马努金的tau函数和素数着色分区。此外，证明了逆贝尔变换能够无缝恢复经典分区递推关系，并提供了一个离散的组合引擎来生成特殊多项式族，包括经典的Appell和Sheffer序列。
Recently, Romik determined in [9] the Taylor expansion of the Jacobi theta constant θ_3, around the point x = 1. He discovered a new integer sequence, (d(n))_0^\infty=1, 1, -1, 51, 849, -26199, \dots, from which the Taylor coefficients are built, and conjectured that the numbers d(n) satisfy certain congruences modulo various primes. In this paper, we prove some of these conjectures, for example that d(n)\equiv (-1)^{n+1}(mod 5) for all n\geq 1,and that for any prime p\equiv 3 (mod 4), d(n) vanishes modulo p for all large enough n.
Some identities on Lin-Peng-Toh's partition statistic of $k$-colored partitions
本文针对Lin-Peng-Toh引入的k-着色分拆统计量$NB_k(r,m,n)$，证明了若干类似于Ramanujan“最美恒等式”的恒等式，并由此重新推出Lin、Peng和Toh先前证明的若干同余式。
The classical congruences satisfied by the Fibonacci and Lucas sequences are reflected with the decomposition of primes in the ring generated by the gold number. This generalizes to establish a correspondence that we hope will be new between Chebotarev's theorem and the congruences satisfied by linear sequences. This link is done into the context of number field extensions. In particular we characterize primes ideals totally decomposed by simple congruences on the terms of linear recurrent sequences. Our results are illustrated by numerous examples, including Padovan sequencesof group $S_3$ an…
A generalization of the Gaussian formula and a q-analog of Fleck's congruence
q-二项式系数是传统二项式系数的多项式表兄弟，并且二项式系数的许多恒等式可以被转换成该多项式设置。例如，帕斯卡三角形的第n行的交替和的熟悉的消失被所谓的高斯公式捕获。在本文中，我们找到了一个q-二项式同余，它综合了这个结果和Fleck的二项式系数的同余。
A Unified Framework for $p$-adic Congruences and $p$-adic Interpolations of Sequences
本文提出一种不依赖p进积分的统一代数框架，用于判定序列的p进同余性与插值函数局部解析性，并将其应用于热诺基数等传统方法无法处理的数论序列。
Combinatorial Properties of Rogers-Ramanujan-Type Identities Arising from Hall-Littlewood Polynomials
在这里，我们考虑$q$ -系列来自霍尔-利特尔伍德多项式，\begin{equation*} R_ν(a,b;q)=\sum_{\substack{λ\\[1pt] λ_1\leq a}} q^{c|λ|} P_{2λ}\big(1,q,q^2,\dots;q^{2b+d}\big). \end{equation*}这些系列是由格里芬，小野，和Warnaar在他们的工作框架的罗杰斯-拉马努金身份。我们设计了一个递归的方法来计算这些系列的系数时，他们出现在罗杰斯-拉马努金框架。此外，我们还研究了这些级数的某些同余式和乘积的同余性质，推广了著名的Ramanujan同余式\begin{equation*} p(5n+4)\equiv0\pmod{5}. \end{equation*}
A generalization of Wolstenholme's harmonic series congruence
给出了Lucas序列的Wolstenholme调和级数同余的一个推广.
Partial Lucas-type congruences
本文证明了一个一般性结果，表明所有可表示为多元Laurent多项式幂的常数项的序列均满足部分Lucas同余，推广了Chamberland和Dilcher关于特定二项和序列模素数同余的研究。
In this paper we present many results and conjectures on congruences involving two types of Apéry-like sequences $\{G_n(x)\}$ and $\{V_n(x)\}$.
The local Dold congruence for Fibonacci and Lucas sequences
本文刻画 Fibonacci 与 Lucas 序列中满足 Dold 同余或可实现性的素数，计算修复乘子并给出四类素数集的密度，分别为 (3/4,1/3,1,1/3) 与 (1/2,1/2,1,1)。
Congruences for sequences analogous to Euler numbers
对于给定的实数$a$，我们通过$E_{0,a}=1$和$E_{n,a}=-a\sum_{k=1}^{[n/2]} \binom n{2k}E_{n-2k,a}$ $(n\ge 1)$定义序列$\{E_{n,a}\}$，其中$[x]$是不超过$x$的最大整数。由于$E_{n,1}=E_n$是第n个欧拉数，$E_{n,a}$可以被看作是欧拉数的自然推广。本文给出了涉及$\{E_{n,a}\}$的一些恒等式和一个反演公式，并建立了$E_{2n,a}\mod{2^{{\rm ord}_2n+8}}$，$E_{2n,a}\pmod{3^{{\rm ord}_3n+5}}$和$E_{2n,a}\pmod{5^{{\rm ord}_5n+4}}$的同余关系，其中${\rm ord}_pn$是使$p^{\a}\mid n$但$p^{\a+1}\nmid n$的最小非负整数$α$.
The Lazard formal group, universal congruences and special values of zeta functions
建立了形式群理论与算术数论之间的联系。特别是，它示出了如何构造与Lazard通用形式群\cite{Tempesta1} - \cite{Tempesta3}相关的通用伯努利多项式的一般Almkvist-Meurman-型同余。它们在$L$ -属乘性序列的理论中的作用是说明。作为一个应用，构造了整数序列。新的同余也得到了，有用的计算特殊值的一类新的Riemann-Hurwitz-型zeta函数。
Reduction on the congruences of partial sums of P-recursive sequences
本文简化了P-递归序列部分和同余式的证明方法，放宽望远镜算子要求，利用序列同余性质，确认了孙的猜想并发现中心三项式系数的新同余式。
Some identities involving associated sequences of special polynomials
本文研究了特殊多项式相伴列的一些性质。从相伴多项式列的性质出发，得到了一些特殊多项式的有趣恒等式。
In this paper, we prove identities for members of the k-generalized Fibonacci sequence with negative indices and we apply these identities to deduce an exact formula for its zero-multiplicity.
Further results on arithmetic properties of biregular overpartitions
本文利用theta函数和分割恒等式，为双正则overpartition的部分同余式提供了初等证明，并提出一种不依赖特定2-分割的模8同余式通用证明方法。
Asymptotic Formulae for Mixed Congruence Stacks
很像重要的工作，哈代和拉马努金证明渐近公式的分拆函数，Auluck和赖特了类似的公式单峰序列。根据Wright的圆方法，我们给出了单峰序列在两参数混合同余关系族上的渐近展开，其中部分在一边直到峰值满足r（mod m），部分在另一边-r（mod m）.在证明中使用的技术包括赖特的循环方法，模块化的转换，并界定复杂的积分。
A partition of $n$ is called a $t$-core partition if none of its hook number is divisible by $t.$ In 2019, Hirschhorn and Sellers \cite{Hirs2019} obtained a parity result for $3$-core partition function $a_3(n)$. Motivated by this result, both the authors \cite{MeherJindal2022} recently proved that for a non-negative integer $α,$ $a_{3^α m}(n)$ is almost always divisible by arbitrary power of $2$ and $3$ and $a_{t}(n)$ is almost always divisible by arbitrary power of $p_i^j,$ where $j$ is a fixed positive integer and $t= p_1^{a_1}p_2^{a_2}\ldots p_m^{a_m}$ with primes $p_i \geq 5.$ In this art…
New representations for all sporadic Apéry-like sequences, with applications to congruences
本文为全部15个散在Apéry型序列构造了基于Laurent多项式常数项的新表示，由此得到不含3或8的幂的二项式表达式，并据此证明超同余式、强Lucas同余及p-进赋值下界。
Some identities arising from Sheffer sequences for the powers of Sheffer pairs under umbral composition
本文研究了阴影复合下Sheffer对幂的Sheffer序列的一些性质。利用这些性质，我们得到了在阴影合成下Sheffer对幂的特殊多项式的Sheffer序列的新的有趣的恒等式。
Congruences Relating Regular Partition Functions, a Genearalised Tau Function and Partition Function Weighted Composition Sums
本文将广义 τ_k(n) 表示为二项式系数加权分拆和，从而建立其与 t-正则分拆函数 R_t(n) 及加权复合和之间的同余恒等式。
Congruences involving binomial coefficients and Lucas sequences
本文得到了中心二项式系数与Lucas序列的一些同余关系。例如，我们证明了如果p>5是素数，则当p= 1，4（mod 5），p= 13，17（mod 30）和p= 7，23（mod 30）时，$\sum_{k=0}^{p-1}F_k*binom(2k,k)/12^k$分别与0，1，-1模p全等，其中{F_n}是Fibonacci序列.我们还提出了几点建议。
Congruences for the Almkvist-Zudilin numbers
给定一个素数$p$，序列$c(n)$整除性的研究有两种方法：$p$ -adic赋值和超同余.对于每个$n$，前者搜索$p$除以$c(n)$的最高幂;而后者（本质上）关注最大幂$r$和$t$，使得$c(p^rn)$与$c(p^{r-1}n)$模$p^t$全等。这就是所谓的超同余。在这篇文章中，我们证明了某些序列的适度超同余，这些序列被称为Almkvist-Zudilin数和另外两个自然相关的序列。
Arithmetic properties of the $\ell$-regular partitions
对于给定的素数$p$，我们分别研究了Ramanujan θ函数$ψ(q)$和$f(-q)$的$p$剖分恒等式的性质.作为应用，我们发现了一些$\ell$ -正则分拆函数，特别是$\ell=2,4,5,8,13,16$的模2同余的无穷族。此外，在Ramanujan给出的$p(n)$的经典同余式的基础上，我们得到了更多的$\ell$ -正则分拆函数的同余式.
Sequences of enumerative geometry: congruences and asymptotics
研究了P^n（n>1）中2n-3次超曲面的直线数的整数序列v_n.我们证明了一些这些数的几种不同类型的同余性质。此外，v_n的渐近性也被描述（在Don Zagier的附录中）。试图在类似的分析其他两个枚举序列：合理的平面曲线的数量和瞬子的数量在五次三倍。
Lucas congruences using modular forms
本文利用模形式证明众多 Apéry 型序列对任意素数满足 Lucas 同余，从而证实 S. Cooper 的四个相关猜想并重新解释已有结果。
On congruence properties of poly-Bernoulli numbers with negative upper-indices
本文研究了负上标多重伯努利数的同余与 $p$ 进性质，并将其结果应用于推导 lonesum 矩阵数量的同余性质。
Congruences and relations for r-Fishburn numbers
最近安德鲁斯和塞勒斯证明了一些惊人的同余式的菲什伯恩号码。我们将他们的结果推广到更一般的数列。结果证明了Fishburn数的一个新的模23同余，并证明了它们对一个相关序列的模5同余。我们还扩展和证明了一些未发表的mattures的Garthwaite和Rhoades。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。