Shashank Kanade, Antun Milas, Matthew C. Russell
We prove certain Nahm-type sum representations for the (odd modulus) Andrews-Gordon identities, the (even modulus) Andrews-Bressoud identities, and Rogers' false theta functions. These identities are motivated on one hand by a recent work of C. Jennings-Shaffer and one of us on double pole series, and, on the other hand, by Córdova, Gaiotto and Shao's work on defect Schur's indices.
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Proofs of five conjectural identities on modular rank four Nahm sums
本文解析证明了Cao和Wang提出的四个关于模秩四Nahm和的猜想恒等式。
Some New Modular Rank Three Nahm Sums from a Lift-Dual Operation
本文通过提升-对偶操作，将Zagier的秩二Nahm和提升至秩三并计算其对偶，证明了所得对偶Nahm和的模性，并建立Rogers--Ramanujan型恒等式将其表示为模无穷乘积。
Some New Modular Rank Four Nahm Sums as Lift-dual of Rank Three Examples
本文通过对Zagier及作者工作中的三阶模性Nahm和应用提升对偶操作，发现九组可能模性的四阶Nahm和，证明其中四组的模性并给出若干猜想恒等式，推进了Nahm和的模性研究。
Some new modular Nahm sums of ranks 3 and 4
通过修改Zagier的秩3例子、对Milas和Wang的秩3蝌蚪Nahm和施以提升对偶运算以及常数项方法，发现了六个新的秩3和秩4的模Nahm和族，并建立了Rogers-Ramanujan型恒等式证明其模性。
Mizuno's rank three Nahm sums I: identities of index $(1,1,2)$
本文确认了Mizuno提出的19个秩三Nahm和的模性，通过建立指标$(1,1,2)$的Rogers--Ramanujan型恒等式，并推广了多重和恒等式。
Nahm sum identities for Cartan matrices of type $D_k$
本文通过贝利配对理论重新证明了与类型 $D_k$ 卡坦矩阵相关的四个纳姆求和恒等式，并给出了其参数推广的两种不同证明。
Remarks on Nahm sums for symmetrizable matrices
本文将Nahm猜想中对称正定矩阵情形下的一个结论推广到可对称化矩阵，证明其模性对应Bloch群元素为挠元，研究了径向渐近性并给出数值候选列表。
Nahm Sums Dual to Zagier's Rank-Three Examples and Related Identities
研究与扎吉尔三阶例子对偶的纳姆和，结合已有工作确立多数对偶的模性。对第九个例子部分证明相关恒等式得条件结果，对第十二个例子给出猜想恒等式。证明依赖特定恒等式，还证明了一些新的纳姆和恒等式及模性。
Proofs of Two Conjectural Identities on Partial Nahm Sums
本文证明 Wang 和 Zeng 关于部分 Nahm 和剩余模族提出的两个 Rogers–Ramanujan 型猜想恒等式，通过 Bailey 对变换公式及两种不同方法将其转化为 Hecke 型级数并最终化为模无穷乘积。
Modularity of tadpole Nahm sums in ranks 4 and 5
本文证明了秩 4 和 5 的蝌蚪 Nahm 和的可模性猜想，并利用降秩公式将其分解为低秩 Nahm 型求和与 theta 函数的混合乘积，同时建立了相应的 Rogers–Ramanujan 型恒等式。
Proofs of Mizuno's Conjectures on Generalized Rank Two Nahm Sums
本研究针对Mizuno提出的14组秩二广义模Nahm和候选集合，通过构造相关Rogers-Ramanujan型恒等式证明了其中8组的模性，给出剩余2组的猜想公式，同时证明了相关向量值函数的模变换猜想，并发现若干新的非模恒等式。
We prove Rogers-Ramanujan type identities for the Nahm sums associated with the tadpole Cartan matrix of rank $3$. These identities reveal the modularity of these sums, and thereby we confirm a conjecture of Penn, Calinescu and the first author in this case. We show that these Nahm sums together with some shifted sums can be combined into a vector-valued modular function on the full modular group. We also present some conjectures for a general rank.
Zagier provided eleven conjectural rank two examples for Nahm's problem. All of them have been proved in the literature except for the fifth example, and there is no $q$-series proof for the tenth example. We prove that the fifth and the tenth examples are in fact equivalent. Then we give a $q$-series proof for the fifth example, which confirms a recent conjecture of Wang. This also serves as the first $q$-series proof for the tenth example, whose explicit form was conjectured by Vlasenko and Zwegers in 2011.
Modularity of Some Nahm Sums as Vector-valued Functions
本文通过将若干族 Nahm 和视为向量值函数建立其模变换公式，证明它们在 $N=1,2,3,4$ 的同余子群 $Γ_0(N)$ 上具有模性，并证明了相关变换公式及 theta 级数的向量值变换公式。
Modular Nahm Sums for the Inverse Cartan Matrix of Type $D_r$
研究针对$D_r$型卡当矩阵逆的模纳姆和，孙和王有相关猜想。通过构造$\lfloor (r + 4)/2\rfloor$个模纳姆和部分证明猜想，利用贝利对方法建立恒等式，证实了猜想中的恒等式。
Proofs of Mizuno's Conjectures on Rank Three Nahm Sums of Index $(1,2,2)$
该研究利用Bailey对理论和$q$-级数技巧，通过建立三重和Rogers--Ramanujan型恒等式，证实了Mizuno提出的除两个外的所有指标为$(1,2,2)$的三阶Nahm和的模性，并证明了两个向量值函数的模变换公式。
Dynkin diagrams, generalized Nahm sums and 2d CFTs
本文探讨广义Nahm和与二维有理共形场论字符的关系，证明某些特定广义Nahm和对应于已知的二维CFTs，如超对称Virasoro最小模型。
The modulo 9 Kanade--Russell identities and their Nahm-sum duals
本文证明了全部五个模9 Kanade--Russell恒等式及其四个广义Nahm和对偶，解决了Wang-Wang和Li-Wang的猜想，并建立了Sun-Wang框架中两个未决情形的模性。
Counterexamples to Zagier's Duality Conjecture on Nahm Sums
针对Nahm问题中Zagier提出的模形式Nahm和的对偶也为模形式的猜想，构造出秩为4的模形式Nahm和反例，其对偶不具备模形式性质，从而证伪该对偶猜想。
Modular Nahm sums for symmetrizable matrices of indices $({2,\ldots, 2},1)$ and $({1,\ldots, 1},2)$
本文提出了针对特定索引的可对称矩阵的模数Nahm和家族，并构建了相关的向量值自动形式。
Bloch torsion and modular combinations of Nahm sums
该研究构造了满足特定条件的12×12矩阵A，证明对应纳姆和非模，得出纳姆猜想中所有解的挠性条件不足以保证模纳姆和存在的结论。
Periodic $Y$-Systems and Nahm Sums: The Rank 2 Case
本文分类了满足辛性质的秩2周期$Y$-系统，发现共有六个，并证明其周期性源于正负向时间演化的变红序列，进而导出量子对数恒等式，同时证实相关Nahm和为模函数。
On a conjecture due to Kanade related to Nahm sums
本文证明了Kanade关于对数函数的猜想，并扩展了两个新的对数恒等式和相关秩-2矩阵。
Asymptotics of parity biases for partitions into distinct parts via Nahm sums
本文推广了Kim等人的奇偶偏差研究，利用圆法给出互异分拆的模N偏差渐近结果，同时提供了分析Nahm型q-超几何级数渐近性的独立有用技术。
Identities on Zagier's rank two examples for Nahm's problem
本文针对纳姆问题中扎吉尔提出的秩二模形式例子，构造罗杰斯-拉马努金型双重求和恒等式，验证了其中十个例子并对其余一个给出猜想。
Rogers--Ramanujan Type Identities for Rank Two Partial Nahm Sums
本文针对秩二部分 Nahm 和，在特定格上找到 14 类对称矩阵使其为模，并建立 Rogers--Ramanujan 型恒等式证明模性。
Machine-Guided Recurrence Boundary Theory for Nahm Sums
该研究开发了机器引导的递推边界理论，结合符号搜索与Q学习发现关键结果，证明了Shi和Wang的猜想3.8，得到Zagier第十二个三阶秩示例对偶纳姆和的二阶递推与五单元证明。
A uniform degree bound for modular rank-two Nahm sums
本文证明秩二对称化 Nahm 和若为模函数，则其 Nahm 点生成域的次数不超过 24，并针对特殊情形给出更紧的界，方法结合牛顿多边形与极点阶分析。
Indefinite theta vectors and generalized Nahm sums of arbitrary rank
本文为任意秩的带符号广义Nahm和构造不定theta表示，确定其模变换法则，并给出相关乘积向量分量的多重和表示。
Two-term dilogarithms, mixed-base Nahm sums, and Fricke symmetry
本文研究两项Rogers二对数关系对秩二Nahm系统的算术约束，确定特定族中的允许步长，证明指标13系统的两项恒等式并精确再现Fricke振幅，同时指出相关乘积求值仍为猜想。
Rank-two Nahm sums, modularity obstructions, and the McKay--Thompson series $T_{31A}$
本文猜想秩二 Nahm 和的正四移位恒等式，通过 Rogers–Ramanujan 函数实现 Monster 级数 $T_{31A}$，证明各分量的非模性并验证组合的渐近行为，提出对偶 Fricke 定律及月光恒等式。
In this note, we report on the results of a computer search performed to find possible modular companions to certain $q$-series identities and conjectures. For the search, we use conditions arising from the asymptotics of Nahm sums. We focus on two sets of identities: Capparelli's identities, and certain partition conjectures made by the author jointly with Matthew C. Russell.
In this paper, we obtain formulas for the number of representations of positive integers as sums of arbitrarily many squares (and other polygonal numbers) with a certain natural weighting. The resulting weighted sums give Fourier coefficients of weight two quasimodular forms.
Sums of the form $\sum_{q \leq N_1 < \cdots < N_m \leq n}{a_{(m);N_m}\cdots a_{(2);N_2}a_{(1);N_1}}$ date back to the sixteen century when Viète illustrated that the relation linking the roots and coefficients of a polynomial had this form. In more recent years, such sums have become increasingly used with a diversity of applications. In this paper, we develop formulae to help with manipulating such sums (which we will refer to as multiple sums). We develop variation formulae that express the variation of multiple sums in terms of lower order multiple sums. Additionally, we derive a set of par…
Rogers-Ramanujan type identities involving double, triple and quadruple sums
本文使用Maple搜索发现了多个涉及二重、三重和四重和的Rogers-Ramanujan型恒等式，并通过直接求和或常数项法将其转化为已知恒等式进行证明，同时得出新的单重和恒等式。
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 $n$.
Some identities of the sums-of-tails type
本文提出一个含参数 b 和 d 的新尾和恒等式，统一了拉马努金的结果，并推导出组合新结果及模拟 theta 函数级数的两个新表示。
Sums of the form $\sum_{N_m=q}^{n}{\cdots \sum_{N_1=q}^{N_2}{a_{(m);N_m}\cdots a_{(1);N_1}}}$ where the $a_{(k);N_k}$'s are same or distinct sequences appear quite often in mathematics. We will refer to them as recurrent sums. In this paper, we introduce a variety of formulas to help manipulate and work with this type of sums. We begin by developing variation formulas that allow the variation of a recurrent sum of order $m$ to be expressed in terms of lower order recurrent sums. We then proceed to derive theorems (which we will call inversion formulas) which show how to interchange the order o…
We consider the reflection identities for harmonic sums at weight four. We decompose a product of two harmonic sums with mixed pole structure into a linear combination of terms each having a pole at either negative or positive values of the argument. The pole decomposition demonstrates how the product of two simpler harmonic sums can build more complicated harmonic sums at higher weight. We list a minimal irreducible bilinear set of reflection identities at weight four which present the main result of the paper. We also discuss how other trilinear and quartic reflection identities can be easil…
Some Generalized Harmonic Number Identities
分部求和用于求包含特定多项式或有理函数的广义调和数的有限级数的和。利用Euler-Maclaurin的幂和公式求出了某些涉及非负整数幂的广义调和数的有限级数的和，并利用该公式求出了涉及多项式的广义调和数的有限级数的和.提供了许多例子和计算机程序。
Fourier transform over finite field and identities between Gauss sums
这是数学的续集AG/0003009本文研究了含有单项有理函数的“指数”的有限域上的“初等函数“的傅里叶变换的恒等式。事实证明，这些身份是由高斯和单项式的身份。我们表明，类似的情况下，复数等身份对应于线性关系某些因子的空间上的乘法字符。
New trigonometric identities and reciprocity laws of generalized Dedekind sums
得到了新的三角恒等式，即余切函数和余割函数的高阶导数的积和式公式。此外，从我们的公式的特殊化，我们不仅得到了各种已知的广义Dedekind和的互易律，而且还得到了新的广义Dedekind和的互易律。
Discovering and Proving Infinite Binomial Sums Identities
我们考虑无穷远处的二项和逆二项和，并将它们重写为一小组常数，例如$π$或$\log(2)$的幂。为了进行这些简化，我们将级数视为生成级数的特化。对于这些生成级数，我们导出了根值迭代积分的整表示。利用代换，我们将积分表示为分圆调和多项式。最后，利用已知的分圆调和多项式之间的关系，我们推导出几个常数的表达式。
Rogers-Ramanujan Type Identities Involving Double Sums
本文利用常数项方法和 Rogers-Szegő 多项式的性质，从经典 Rogers-Ramanujan 恒等式出发，证明了四个新的双重级数 Rogers-Ramanujan 型恒等式。
We establish Ohno-type identities for multiple harmonic ($q$-)sums which generalize Hoffman's identity and Bradley's identity. Our result leads to a new proof of the Ohno-type relation for $\mathcal{A}$-finite multiple zeta values recently proved by Hirose, Imatomi, Murahara and Saito. As a further application, we give certain sum formulas for $\mathcal{A}_2$- or $\mathcal{A}_3$-finite multiple zeta values.
Indefinite theta functions arising from affine Lie superalgebras and sums of triangular numbers
本文通过推广 Roehrig 和 Zwegers 的不定 theta 函数理论，证明若干幂级数为模形式，并由此导出三角数生成函数幂的恒等式，进而揭示其与仿射李超代数分母恒等式的联系。
Dedekind sums occur in the transformation behaviour of the logarithm of the Dedekind eta-function under substitutions from the modular group. In 1892, Dedekind showed a reciprocity relation for the Dedekind sums. Apostol generalized Dedekind sums by replacing the first Bernoulli function appearing in them by any Bernoulli functions and derived a reciprocity relation for the generalized Dedekind sums. In this paper, we consider poly-Dedekind sums which are obtained from the Dedekind sums by replacing the first Bernoulli function by any type 2 poly-Bernoulli functions of arbitrary indices and pr…
On new identities of Jacobi sums and related cyclotomic matrices
本文利用Jacobi和的算术性质，研究涉及Jacobi和的乘积，并揭示这些乘积与特定循环矩阵之间的联系，验证了Sun在2019年提出的猜想并获得更强的结果。
Identities involving cyclic sums of regularized multiple zeta values each of depth less than $5$
本文给出了深度小于$5$的正则化重zeta值的循环和的恒等式。作为推论，我们提出了两种推广的霍夫曼定理的对称和多个zeta值的情况下。
In a one-page fragment published with his lost notebook, Ramanujan stated two double series identities associated, respectively, with the famous Gauss Circle and Dirichlet Divisor problems. The identities contain an "extra" parameter, and it is possible that Ramanujan derived these identities with the intent of attacking these famous problems. Similar famous unsolved problems are connected with $f_K(n)$, the number of integral ideals of norm $n$ in an algebraic number field $K$. In this paper we establish Riesz sum identities containing an "extra" parameter and involving $f_K(n)$, or divisor f…
The class of Lambert series generating functions (LGFs) denoted by $L_α(q)$ formally enumerate the generalized sum-of-divisors functions, $σ_α(n) = \sum_{d|n} d^α$, for all integers $n \geq 1$ and fixed real-valued parameters $α\geq 0$. We prove new formulas expanding the higher-order derivatives of these LGFs. The results we obtain are combined to express new identities expanding the generalized sum-of-divisors functions. These new identities are expanded in the form of sums of polynomially scaled multiples of a related class of divisor sums depending on $n$ and $α$.
New trigonometric identities and reciprocity laws of generalized Dedekind sums II
我们得到了新的三角恒等式，它们是余割函数和余切函数的导数的积和型公式。此外，从我们的公式的特殊化，我们得到新的互惠法律的广义Dedekind和。
Symmetric identities for Euler polynomials
本文建立了关于欧拉多项式乘积和的两个对称恒等式。
Trigonometric analogue of the identities associated with twisted sums of divisor functions
本文受 Ramanujan 丢失笔记启发，推导出与除数函数扭曲和相关恒等式的三角函数等价版本，并给出了 $r_6(n)$ 的模拟恒等式。
A Generalization of the Doubling Construction for Sums of Squares Identities
加倍构造法是从已知恒等式的平方和中生成Hurwitz问题新解的一种快速而重要的方法。本文推广了加倍构造，并从任意给定的容许三元组$[r,s,n]$中得到了一系列新的构造$[r+ρ(2^{m-1}),2^ms,2^mn]$，其中$ρ$是Hurwitz-Radon函数，对所有正整数$m$.
We show that identities involving trigonometric sums recently proved by Harshitha, Vasuki and Yathirajsharma, using Ramanujan's theory of theta functions, were either already in the literature or can be proved easily by adapting results that can be found in the literature. Also we prove two conjectures given in that paper. After mentioning many other works dealing with identities for various trigonometric sums, we end this paper by describing an automated approach for proving such trigonometric identities.
Some identities involving Prouhet-Thue-Morse sequence and its relatives
令$s_{k}(n)$表示整数$n$在基$k$中的位数之和。出于某些身份的Nieto，贝特曼和布拉德利涉及的形式$\sum_{i=0}^{2^{n}-1}(-1)^{s_{2}(i)}(x+i)^{m}$ $m=n$和$m=n+1$的总和，我们认为序列的多项式\begin{equation*} f_{m,n}^{\mathbf u}(x)=\sum_{i=0}^{k^{n}-1}ζ_{k}^{s_{k}(i)}(x+{\mathbf u}(i))^{m}. \end{equation*}定义的序列${\bf u}(i)$满足一定的递归关系。我们证明，计算这些多项式是计算其常数项基本上是等价的，我们找到了一个明确的公式，这个数字。这使我们能够证明几个有趣的身份涉及总和的二进制数字。我们还证明了一些相关的结果，这些结果是独立的，可以看作是某些涉及Prouhet-Thue-Morse序列和的进一步推广。
New Multiple Harmonic Sum Identities
考虑一类特殊的含有调和数的二项和，利用部分分式分解的初等方法证明了三个恒等式。给出了它在无穷级数和同余式中的一些应用。
Some $q$-analogs of congruences for central binomial sums
我们建立了四个包含中心二项式系数的同余式的$q$ -模拟。通过$q$ -WZ方法显示了为此目的所需的$q$ -恒等式。
Relations between Chebyshev, Fibonacci and Lucas polynomials via trigonometric sums
本文通过有限三角和建立斐波那契、卢卡斯与切比雪夫多项式间的新恒等式，并给出其物理解释及与图有效电阻的联系。
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