$q$-Identities related to overpartitions and divisor functions
Amy M. Fu, Alain Lascoux
我们推广并证明了Corteel和Lovejoy关于超划分和除数函数的定理。
We generalize and prove conjectures of Corteel and Lovejoy, related to overpartitions and divisor functions.
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Overpartitions, lattice paths and Rogers-Ramanujan identities
我们扩展分区理论工作的安德鲁斯，Bressoud和伯奇overpartitions，定义的概念，连续的行列，广义Durfee平方和广义格路径，然后将这些overpartitions定义的多重条件的部分。这导致了许多新的划分和过度划分的身份，并提供了一个统一的一些著名的身份的罗杰斯-拉马努金类型。其中包括Gordon对Rogers-Ramanujan恒等式的推广，Andrews对Göllnitz-Gordon恒等式的推广，以及Lovejoy的"Gordon's overpartitions定理。"
Bressoud Style Identities for Regular Partitions and Overpartitions
我们构建了一个家庭的分区身份，其中包含以下身份：罗杰斯-Ramanujan-戈登身份，Bressoud的甚至模推广他们，以及他们的同行overpartitions由于Lovejoy等人。和陈等人。我们获得不寻常的同伴身份已知定理以及新的过程中。这种证明是与传统相反的，是建设性的，并且对自动化开放。
Overpartitions Identities Involving Gaps and Weights
在本文中，我们找到了一个恒等式，它将超划分函数与Roger-Ramanujan-Gordon型超划分函数联系起来，并考虑了权和间隙。这个恒等式可以看作是Alladi加权恒等式的一个类似物。
Sang, Shi and Yee, in 2020, found overpartition analogs of Andrews' results involving parity in Rogers-Ramanujan-Gordon identities. Their result partially answered an open question of Andrews'. The open question was to involve parity in overpartition identities. We extend Sang, Shi, and Yee's work to arbitrary moduli, and also provide a missing case in their identities. We also unify proofs of Rogers-Ramanujan-Gordon identities for overpartitions due to Lovejoy and Chen et.al.; Sang, Shi, and Yee's results; and ours. Although verification type proofs are given for brevity, a construction of se…
The combinatorics of identities involving overpartitions with distinct parts
该研究针对Andrews与EI Bachraoui提出的涉及分拆和不同部分超分拆的$q$-级数恒等式，按分拆部分数细化相关结果并给出组合证明。
Combinatorial Proofs for Two-Color Partition Identities Involving Overpartitions and Minimal Excludants
研究涉及超分割和最小排除数的双色分拆恒等式，通过构造显式保权双射和奇偶反转对合给出组合证明，还给出相关超分割精细化的新组合证明。
In this paper, we combined two types of partitions and introduced 2-colored Rogers-Ramanujan partitions. By finding some functional equations and using a constructive method, some identities have been found. Some Overpartition identities coincide with our findings. A correspondence between colored partitions and overpartitions is provided.
A Combinatorial Proof of an Identity of Malik and Sarma on Overpartitions with Repeated Smallest Non-Overlined Part
本文为Malik和Sarma提出的关于具有重复最小非上划线部分的超分拆的四个恒等式中剩余的一个提供了组合证明，解决了该开放问题。
On a dual and an overpartition generalization of a family of identities of Andrews
我们提出了一个双重的一个家庭的分区身份的安德鲁斯涉及分区没有重复的奇数部分（除其他条件），以及overpartition概括，封装两个家庭。这些是在作者与Debajyoti Nandi一起为即将发表的一篇文章进行研究的过程中发现的。证明使用Appell的比较定理。
In 1980, Bressoud conjectured a combinatorial identity $A_j=B_j$ for $j=0$ or $1$, where the function $A_j$ counts the number of partitions with certain congruence conditions and the function $B_j$ counts the number of partitions with certain difference conditions. Bressoud's conjecture specializes to a wide variety of well-known theorems in the theory of partitions. Special cases of his conjecture have been subsequently proved by Bressoud, Andrews, Kim and Yee. Recently, Kim resolved Bressoud's conjecture for the case $j=1$. In this paper, we introduce a new partition function $\bar{B}_j$ whi…
Combinatorial proofs of some identities on overpartitions with repeated smallest non-overlined part
本文通过组合方法证明了关于带有重复最小非重线部分的划分恒等式。
Overpartitions with parts separated by parity
本文研究了具有奇偶性间隔部分的超分拆的生成函数，并证明了与模形式、q-超几何级数和拟模形式相关的q-级数恒等式。
We obtain a three-parameter $q$-series identity that generalizes two results of Chan and Mao. By specializing our identity, we derive new results of combinatorial significance in connection with $N(r, s, m, n)$, a function counting certain overpartition pairs recently introduced by Bringmann, Lovejoy and Osburn. For example, one of our identities gives a closed-form evaluation of a double series in terms of Chebyshev polynomials of the second kind, thereby resulting in an analogue of Euler's pentagonal number theorem. Another of our results expresses a multi-sum involving $N(r, s, m, n)$ in te…
An Overpartition Companion of Andrews and Keith's 2-colored $q$-series Identity
本文推广 Andrews 和 Keith 的 Schmidt 型划分定理，类比 Franklin 推广 Glaisher 定理，得到超划分情形下 2-着色恒等式的伴随恒等式，并指出其可能为更一般结果的特例。
Some further q-series identities related to divisor functions
我们给出了Dilcher和Schlinger关于除数函数的一些q-级数恒等式的新的推广。一些有趣的特殊情况下，也推导出，包括一个身份有关的研究Corteel和Lovejoy的超分割。
Anti-lecture Hall Compositions and Overpartitions
我们表明，反讲堂组成的第一个条目不超过k-2的n等于overpartition的数量n与非overlined部分不全等的$0,\pm 1$模k。这个恒等式可以被认为是Corteel和Savage的反讲堂定理的改进版本。为了证明这一结果，我们找到了两个Rogers-Ramanujan型恒等式，它们与Andrews的Rogers-Ramanjan型恒等式类似。当k是奇数时，我们给出了另一种证明，通过使用广义罗杰斯-拉马努金身份由于安德鲁斯，一个双射的Corteel和萨维奇和一个改进版本的双射也由于Corteel和萨维奇。
Overpartition pairs and two classes of basic hypergeometric series
研究了两类基本超几何级数的组合数学。我们首先证明这些级数是由零件频率条件定义的某些过划分对的母函数。然后，我们表明，当专门这些系列也是生成函数的overpartition对有界连续的行列，overpartition对的条件，其Durfee解剖，以及某些格路径。当进一步专门化时，级数变成无穷乘积，导致划分、超划分和超划分对的许多恒等式。
Rank deviations for overpartitions
本文证明了两个overpartition秩偏离平均值的一般公式，该公式以Appell--Lerch级数和theta函数的商之和表示，可用于恢复文献中的overpartition秩差分恒等式，并给出了两个说明。
An overpartition analogue of $q$-binomial coefficients, II: combinatorial proofs and $(q,t)$-log concavity
在以前的论文中，我们研究了高斯多项式的过分割模拟作为过分割拟合在$m \times n$矩形内的生成函数。在这里，我们增加了一个参数计数的数量overlined部分，获得一个两个参数的推广$\overline{m+n \brack n}_{q,t}$的高斯多项式，这也是一个$(q,t)$ -模拟德兰诺伊数。首先，我们得到有限版本的经典$q$ -系列的身份，如$q$ -二项式定理和勒贝格身份，以及两个变量的推广的经典身份涉及高斯多项式。然后，通过构造对合，得到了一个涉及有限θ函数的恒等式，并证明了$\overline{m+n \brack n}_{q,t}$的$(q,t)$ -log恒等式.我们特别强调的作用，组合证明和后果，我们的结果Delannoy数。最后，我们对$\overline{m+n \brack n}_{q,t}$的单峰性进行了一些猜测。
On some $q$-identities related to divisor functions
我们给出了Dilcher，Fu和Lascoux关于除数函数的$q$恒等式的推广和简单证明.
On q-series identities with multiple sums related to divisor functions
本文系统阐述多组与除数函数相关的多重和q-恒等式，并针对每个恒等式推导出更强的结果。
On an Overpartition Analogue of $SOME(n)$
本文研究了划分类比的过划分函数，推导了生成函数并获得模3、5及2的幂的同余式。
This paper is concerned with a class of partition functions $a(n)$ introduced by Radu and defined in terms of eta-quotients. By utilizing the transformation laws of Newman, Schoeneberg and Robins, and Radu's algorithms, we present an algorithm to find Ramanujan-type identities for $a(mn+t)$. While this algorithm is not guaranteed to succeed, it applies to many cases. For example, we deduce a witness identity for $p(11n+6)$ with integer coefficients. Our algorithm also leads to Ramanujan-type identities for the overpartition functions $\overline{p}(5n+2)$ and $\overline{p}(5n+3)$ and Andrews--P…
Moments of the Crank Statistic for $t$-Core Partitions and Overpartitions
本文利用统一矩-迹框架，对$t$-核分拆和超分拆的曲柄统计量建立迹与逆迹公式，并借助完全Bell多项式导出分拆数的显式表达式。
Universal mock theta functions and two-variable Hecke-Rogers identities
得到了三个泛模拟θ函数的二元Hecke-Rogers恒等式.这意味着许多拉马努金的模拟θ函数，包括所有的三阶函数，有一个赫克-罗杰斯型双和表示。我们发现了Dyson秩函数、过划分秩函数、M2秩函数和相关的spt-crank函数的新的母函数恒等式。利用基本超几何函数理论证明了结果。
Two truncated identities of Gauss
给出了高斯三角级数与平方级数部分和的两个新的展开式。因此，我们得到了一族不等式的过划分函数$\bar{p}(n)$和分拆函数$p_1(n)$计数分区的$n$不同的奇数部分。作为证明，进一步提出了关于分拆函数变分的不等式.
Using basic hypergeometric functions and partial fraction decomposition we give a new kind of generalization of identities due to Uchimura, Dilcher, Van Hamme, Prodinger, and Chen-Fu related to divisor functions. An identity relating Lambert series to Eulerian polynomials is also proved.
Arithmetic properties of MacMahon-type sums of divisors
本文证明了MacMahon广义除数和函数扩展的生成函数$U_t(a,q)$系数满足新的Ramanujan型同余无限族，并得出关于几乎3-regular overpartitions数量的同余性质。
Restricted k-color partitions
我们将overpartitions推广到（k，j）-colored partitions：k-colored partitions，其中每个部分大小最多可以有j种颜色。我们发现了许多同余和其他对称性。我们使用各种各样的工具来证明我们的定理：生成函数剖分、模形式、双射和其他组合映射。我们发现连接到除数和，韩/Nekrasov-Okounkov钩长公式和一个可能的方法，有限化，和其他主题，表明丰富的矿山的结果是可用的。
In this article we extend a theorem of Andrews, Crippa, and Simon on the asymptotic behavior of polynomials defined by a general class of recursive equations. Here the polynomials are in the variable $q$, and the recursive definition at step $n$ introduces a polynomial in $n$. Our extension replaces the polynomial in $n$ with either an exponential or periodic function of $n$.
In this article, a $q$-series examined by Kluyver and Uchimura is generalized. This allows us to find generalization of the identities in the random acyclic digraph studied by Simon, Crippa, and Collenberg in 1993. As one of the corollaries of our main theorem, we get results of Dilcher and Andrews, Crippa, and Simon. This main theorem involves a surprising new generalization of the divisor function $σ_s(n)$, which we denote by $σ_{s,z}(n)$. Analytic properties of $σ_{s,z}(n)$ are also studied. As a special case of one of our theorem we obtain result from a recent paper of Bringmann and Jennin…
In 1984, Bressoud and Subbarao obtained an interesting weighted partition identity for a generalized divisor function, by means of combinatorial arguments. Recently, the last three named authors found an analytic proof of the aforementioned identity of Bressoud and Subbarao starting from a $q$-series identity of Ramanujan. In the present paper, we revisit the combinatorial arguments of Bressoud and Subbarao, and derive a more general weighted partition identity. Furthermore, with the help of a fractional differential operator, we establish a few more Bressoud-Subbarao type weighted partition i…
George Andrews and Ae Ja Yee recently established beautiful results involving bivariate generalizations of the third order mock theta functions $ω(q)$ and $ν(q)$, thereby extending their earlier results with the second author. Generalizing the Andrews-Yee identities for trivariate generalizations of these mock theta functions remained a mystery, as pointed out by Li and Yang in their recent work. We partially solve this problem and generalize these identities. Several new as well as well-known results are derived. For example, one of our two main theorems gives, as a corollary, a special case …
Using a specific form of the triple product identity, polygonal number identities are stated. Further number identities are examined that can be considered identities related to modular sets of numbers. The identities can be used to give results on integer partitions with parts from numbers in modular arithmetic progression. This includes recursive formulas for the number of partitions using these modular parts. The triple product identity can derive further recursive formulas. Additionally, there is a recursive formula for the related sum of divisors function. The specific triple product iden…
The method of weighted words revisited
阿拉迪和戈登介绍了方法的加权词在1993年证明了一个完善和推广舒尔的分区身份。后来，他们和安德鲁斯一起用它来完善卡帕雷利和格尔尼茨的身份。在本文中，我们提出了一个新的变种，这种方法，它可以用来研究更复杂的划分身份，并应用它来证明三个划分身份的细化和推广。第一个是Siladić定理（2002），来自于顶点算子代数。第二个是Primc（1999）的结构恒等式，它来自于晶体基础理论.最后一个是一个非常普遍的身份有色overpartitions概括和统一的几个概括舒尔定理由于阿拉迪，戈登，安德鲁斯，Corteel-Lovejoy，Lovejoy和作者。
Uchimura, in 1987, introduced a probability generating function for a random variable $X$ and using properties of this function he discovered an interesting $q$-series identity. He further showed that the $m$-th cumulant with respect to the random variable $X$ is nothing but the generating function for the generalized divisor function $σ_{m-1}(n)$. Simon, Crippa, and Collenberg, in 1993, explored the $G_{n,p}$-model of a random acyclic digraph and defined a random variable $γ_n^{*}(1)$. Quite interestingly, they found links between limit of its mean and the generating function for the divisor …
Properties of the recursive divisor function and the number of ordered factorizations
本文研究递归除数函数 $κ_x(n)$，给出其 Dirichlet 级数、与普通除数函数的卷积关系及若干算术恒等式，并阐明其与有序因子分解数 $K(n)$ 的联系。
A Combinatorial Proof of Partition Identities for the Number of Distinct Part Sizes
本文对Bressoud等提出的加权分拆恒等式给出组合证明，并将结果扩展到p⁽ᵏ⁾(n)，即n恰好拆分为k个不同分拆大小的分拆数。
Extensions of MacMahon's sums of divisors
本文推广MacMahon的除数和概念，通过有理函数逼近MacMahon型生成函数，揭示了丰富的整除性定理和新的组合恒等式，并给出多重q-调和和的实例。
Duality for Finite Multiple Harmonic q-Series
我们定义了两个有限的q-类似物的某些多重调和级数与任意数量的自由参数，并证明了这些q-类似物的身份，表示他们在涉及高斯二项式系数的多重嵌套和。这些恒等式的特殊情况--例如，所有参数都等于1--在文献中出现过。只有一个参数的特殊情况下减少到一个单位的除数生成函数，这已经收到了一些关注与排序理论中的问题。一般情况下，可以被视为一个对偶结果，让人联想到普通的多重zeta函数的对偶关系。
We study the $q$-bracket operator of Bloch and Okounkov when applied to $f(λ)=\sum_{λ_i \in λ}g(λ_i)$ and $f(λ)=\sum_{\substack{λ_i \in λλ_i \text{distinct} }}g(λ_i)$. We use these expansions to derive convolution identities for the functions $f$ and link both classes of $q$-brackets through divisor sums. As a result, we generalize Euler's classic convolution identity for the partition function and obtain an analogous identity for the totient function. As corollaries, we generalize Stanley's theorem as well as provide several new combinatorial results.
Closed-Form Formula for the Partition Function and Related Functions
本文为分拆函数及 Andrews 的 spt 函数推广提出了新的闭式算术与递归公式，利用容斥原理推导了非互质分拆函数公式，并通过分拆的最大公约数定理将分拆与素数联系，推导出互质函数公式及多个数论函数的新恒等式。
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