Alexander E Patkowski
We prove an interesting symmetric $q$-series identity which generalizes a result due to Ramanujan. A proof that is analytic in nature is offered, and a bijective-type proof is also outlined.
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
An identity involving symmetric sums of regularized multiple zeta-star values of harmonic type was proved by Hoffman. In this paper, we prove an identity of shuffle type. We use Bell polynomials appearing in the study of set partitions to prove the identity.
An identity connecting theta series associated with binary quadratic forms of discriminant $Δ$ and $Δ($prime$)^2$
本文陈述并证明了一个恒等式，它连接了与二元二次型的恒等判别式$Δ$和$Δp^2$有关的θ级数，其中$p$是素数.利用这个恒等式，我们将判别式$Δp^2$的θ级数写成Lambert级数的线性组合，从而推广了Toh的结果.然后，我们使用这些朗伯级数分解给出明确的表示公式的形式的判别式$Δp^2$。最后，我们给出了我们的主要身份，它采用了Buell的映射连接形式的判别$Δ$到$Δp^2$的推广。我们的广义恒等式将与单一形式的判别式$Δ$相关联的θ序列链接到与形式的判别式$Δp^2$相关联的θ序列，其中$Δ$和$Δp^2$不再需要是恒等式。
Inspired by a famous identity of Ramanujan, we propose a general formula linearizing the convolution of Dirichlet series as the sum of Dirichlet series with modified weights; its specialization produces new identities and recovers several identities derived earlier in the literature, such as the convolution of squares of Bernoulli numbers by A. Dixit and collaborators, or the convolution of Bernoulli numbers by Y. Komori and collaborators.
We present a new "integral=series" type identity of multiple zeta values, and show that this is equivalent in a suitable sense to the fundamental theorem of regularization. We conjecture that this identity is enough to describe all linear relations of multiple zeta values over Q. We also establish the regularization theorem for multiple zeta-star values, which too is equivalent to our new identity. A connection to Kawashima's relation is discussed as well.
When the discriminants $Δ$ and $Δp^2$ are idoneal, Patane proved a theorem which connects the theta series associated to binary quadratic forms of each discriminant. This paper generalizes the main theorem of Patane by no longer requiring $Δ$ and $Δp^2$ to be idoneal. In particular, we state and prove an identity which connects the theta series associated to a single binary quadratic form of discriminant $Δ$ to a theta series associated to a subset of binary quadratic forms of discriminant $Δp^2$. Here and everywhere $p$ is a prime.
A generating function identity for finite and symmetric multiple zeta values via Dougall's bilateral ${}_2H_2$ summation
本研究利用Dougall双边${}_2H_2$求和公式建立有限与对称多重zeta值的生成函数恒等式，应用中证明指标分量奇偶性受限的和式公式，还推导有限及对称多重zeta值的生成函数恒等式，证明有限多重zeta值的Kaneko猜想及其对称类似。
A Product Identity For Dirichlet Series Satisfying Hecke's Functional Equation
本文为满足Hecke函数方程的Dirichlet级数建立了Wilton乘积公式的类似物，并应用于多种L-函数与zeta函数，还给出了Riemann zeta函数的4项乘积恒等式。
We show that a binomial identity arising in the context of the study of series expansions of $1/π$ can be seen as an incarnation of Whipples second theorem for hypergeometric series.
An elementary proof of a power series identity for the weighted sum of all finite abelian p-groups
利用组合技巧，证明了所有有限交换$p$ -群$\sum_G |G|^{-u} |\text{Aut}(G)|^{-1}$的自同构的倒数的加权和等于$\prod_{j=u+1}^\infty\left(1-1/p^j\right)^{-1}$，其中$u$是非负整数.这个结果最初是由H. Cohen和H. W.小伦斯特拉本文给出了其结果的一个新的初等证明。
A Comment on Matiyasevich's Identity #0102 with Bernoulli Numbers
本文将Matiyasevich的恒等式#0102与Bernoulli数以及Candelpergher，Coppo和Delabaere关于调和数无穷和的发散级数的Ramanujan求和的恒等式联系起来并加以推广.这些公式是Euler和的解析延拓，并导出了Bernoulli数导数的新的递推关系.所使用的技术是围道积分，生成函数和发散级数。
An Extension of the Classical Gauss Series-product Identity by Fermionic Construction of \hat{sl}_n
主要结果是基于经典Gauss恒等式和仿射李代数的不可约最高权模的特征标公式的两种不同解释，得到了两类无穷级数积恒等式.
On a certain identity for the cotangent zeta finite Dirichlet series and its application to the Berndt--Arakawa formula
本文通过建立两个有限余切 Dirichlet 级数的渐近恒等式，为 Berndt–Arakawa 公式提供了一种新颖简洁的证明。
In this paper we give a convolution identity for the complete and elementary symmetric functions. This result can be used to proving and discovering some combinatorial identities involving $r$-Stirling numbers, $r$-Whitney numbers and $q$-binomial coefficients. As a corollary we derive a generalization of the quantum Vandermonde's convolution identity.
An identity for generating series of deformations of multiple zeta values within an algebraic framework
本文在代数框架内推广Bachmann恒等式，证明多重zeta值变形生成级数的一般恒等式，并得到含Lewis-Zagier周期类似函数的新公式。
Applying the theory of modular forms and Lambert series manipulations we establish an Eisenstein series identity. From this formula we confirm a Lambert series identity conjectured by Gosper. Another Lambert series identity of Gosper is also confirmed by using Lambert series manipulations.
A doubly infinite set of series expansion for $1/π$ are reported. They follow trivially from a formal expansion for the quotient of the values taken by the gamma function for two (complex) arguments differing by an integer plus one half, obtained by an alternative computation of the Wronskian of the modified Bessel functions. The same formal expansion allows to discover also a new binomial identity.
A spectral identity for second moments of Eisenstein series of O(n, 1)
设H=O（n）xO（1）是G=O（n，1）的各向异性子群，A是k=Q的Adele环.考虑G上的爱森斯坦级数对H上的形式F的周期。依赖于一个变体的索伯列夫空间，我们描述了某些庞加莱级数作为基本解的拉普拉斯算子，并使用它们来建立一个谱恒等式有关的二阶矩（在F方面）的爱森斯坦级数。
An unpublished identity of Gosper restates a hypergeometric identity for odd zeta values in terms of an infinite product of matrices. We show that this correspondence runs much deeper, and show that many examples of WZ-accelerated series for zeta values lift to infinite matrix products. We also introduce a new matrix subgroup, the Gosper group, which all of our matrix products fall into.
Bhoria, Eyyunni and Maji recently obtained a four-parameter $q$-series identity which gives as special cases not only all five entries of Ramanujan on pages 354 and 355 of his second notebook but also allows them to obtain an analytical proof of a result of Bressoud and Subbarao. Here, we obtain a finite analogue of their identity which naturally gives finite analogues of Ramanujan's results. Using one of these finite analogues, we deduce an identity for a finite sum involving a ${}_2ϕ_1$. This identity is then applied to obtain a generalization of the generating function version of Andrews' f…
We give a general identity relating Eisenstein series on general linear groups. We do it by constructing an Eisenstein series, attached to a maximal parabolic subgroup and a pair of representations, one cuspidal and the other a character, and express it in terms of a degenerate Eisenstein series. In the local fields analogue, we prove the convergence in a half plane of the local integrals, and their meromorphic continuation. In addition, we find that the unramified calculation gives the Godement-Jacquet zeta function. This realizes and generalizes the construction proposed by Ginzburg and Soud…
An identity of Andrews, multiple integrals, and very-well-poised hypergeometric series
本文给出了Zudilin的一个定理的新证明，该定理等价于一个非常平衡的超几何级数和一个特殊的多重积分。这个积分推广了Vasilenko和Vasilyev的积分，这些积分被提出作为研究黎曼ζ函数值在整数上的算术行为的工具。我们的证明是基于极限情况下的一个基本超几何身份的安德鲁斯。
Truncated Theta Series Related to the Jacobi Triple Product Identity
本文证明了Merca关于雅可比三重积截断恒等式猜想的前三种情况及相关恒等式，提出新组合解释，并用组合方法证明了可简化为已知猜想的恒等式。
Beyond endoscopy for the Symmetric Cube L-function
本文是第一次尝试获得信息的对称功率表示的$GL_2$自守形式通过一个跟踪公式，这是超越内窥镜技术。特别地，我们研究了所有形式的对称三次幂表示在$\Q$邻接的单位立方根。在这项研究中所需要的一个关键工具是一个身份相关的三次指数和Kloosterman和。同样的身份是至关重要的基本引理的跟踪公式比较工作的毛泽东和拉利斯。
A generalization of a beautiful $q$-series identity found in the unorganized portion of Ramanujan's second and third notebooks is obtained. As a consequence, we derive a new three-parameter identity which is a rich source of partition-theoretic information. In particular, we use this identity to obtain a generalization of a recent result of Andrews, Garvan and Liang, which itself generalizes the famous result of Fokkink, Fokkink and Wang. This three-parameter identity also leads to several new weighted partition identities as well as a natural proof of a recent result of Garvan. This natural p…
The Euler Series Transformation and the Binomial Identities of Ljunggren, Munarini and Simons
它表明，好奇的身份西蒙斯如下立即从欧拉的级数变换公式，也从一个身份由于Ljunggren。本文还讨论了西蒙斯恒等式与勒让德多项式的关系。最后利用广义欧拉级数变换得到了Munarini的两个最近的二项式恒等式。
A new four parameter q-series identity and its partition implications
我们证明了一个新的四参数q-超几何级数的身份，从三个参数的关键身份的Goellnitz定理由于Alladi，安德鲁斯和戈登，以下作为一个特殊情况下，设置一个参数等于0。新恒等式等价于一个四参数分割定理，它推广了Goellnitz的深定理，从而解决了Andrews在30年前提出的一个问题。最后，对Jacobi三重积恒等式的四重积推广等结果进行了简要的讨论，并对今后的研究进行了展望。
A Weighted Sum Formula for Double Eisenstein Series
研究双艾森斯坦级数的加权和公式，通过作者与田坂证明的受限双洗牌关系得出双艾森斯坦级数恒等式，用组合方法及生成级数证明除数和恒等式，实现对相关恒等式的证明。
In this paper we give a new proof of the Ohno-Nakagawa Theorem using the techniques of $L$-series. By applying Eisenstein's parametrization of binary cubic forms on the one hand, and a class field theory interpretation of Datskovsky \& Wright's Theorem on the other, we reduce the Ohno-Nakagawa Theorem to an identity involving the $L$-series and the truncated $L$-series of quadratic orders. We prove this identity by establishing a general relation among these two types $L$-series.
q-series and L-functions related to half-derivatives of the Andrews--Gordon identity
研究了Zagier的q-级数恒等式的推广。引入了一个非正整数的L函数的母函数，它被看作Andrews-Gordon q-级数的半微分.当q是单位单位根时，生成函数与环面结的量子不变量一致。
Proofs of conjectures on Ramanujan-type series of level 3
本文针对最具挑战性的3级拉马努金型级数，借助椭圆函数与Kummer-Goursat变换极限情形提出新恒等式，证明了三个已有的猜想公式并发现一个新的四次无理数参数的3级拉马努金型级数。
A $q$-analog of the Bailey-Borwein-Bradley identity
我们建立了一个$q$ -模拟的Bailey-Borwein-Bradley恒等式生成加速系列甚至zeta值和证明$q$ -模拟的马尔可夫和Amdeberhan的系列$ζ(3)$使用$q$ -Markov-WZ方法。
Ramanujan recorded five interesting q-series identities in a section that is not as systematically arranged as the other chapters of his second notebook. These five identities do not seem to have acquired enough attention. Recently, Dixit and the third author found a one-variable generalization of one of the aforementioned five identities. From their generalized identity, they were able to derive the last three of these q-series identities, but didn't establish the first two. In the present article, we derive a one-variable generalization of the main identity of Dixit and the third author from…
Symmetric identities on Bernoulli polynomials
本文得到了Carlitz关于伯努利多项式的一个恒等式的推广。然后我们利用这个推广公式导出了两个对称恒等式，这两个恒等式可归结为关于伯努利多项式和伯努利数的一些已知恒等式，包括Miki恒等式.
The purpose of this note is to offer some partition implications of a $q$-series that is connected to the second Atkin-Garvan moment. Inequalities and relationships among the number of divisors and partitions are provided as consequences.
We prove a double-sum analog of an identity known to Kronecker and then express it in terms of functions studied by Appell and Kronecker's student Lerch, in so doing we show that the double-sum analog is of mixed mock modular form. We also give related symmetric generalizations.
We will prove an identity involving refined $q$-trinomial coefficients. We then extend this identity to two infinite families of doubly bounded polynomial identities using transformation properties of the refined $q$-trinomials in an iterative fashion in the spirit of Bailey chains. One of these two hierarchies contains an identity which is equivalent to Capparelli's first Partition Theorem.
An Arithmetic Transfer Identity
我们证明了张伟的算术基本引理猜想的一个变种，其中n=2。更确切地说，我们考虑的变形长度的某些准同态的准正则升降机的意义下的格罗斯。本文证明了在与GL_2有关的对称空间上存在一个检验函数，它在GL_1上的轨道积分等于所讨论的变形长度。
Cubic identities for theta series in three variables
我们介绍了三个变量的类似物的θ系列的Borwein和Borwein。我们证明了各种身份涉及这些θ系列包括一个概括的三次身份的Borwein和Borwein。
We develop a new theory of $L$-series based on replacing Dirichlet characters mod $N$ by symmetric functions mod $N$ in order to calculate explicitly the sums of infinite series more closely related to $ζ(2n+1)$, specifically $ζ(3)$, $ζ(5)$. This generalizes the corresponding theory of sums of $L$-series associated to Dirichlet characters.
An Explicit Cohen-Style Threefield Identity
本文基于Andrews等和Cohen的工作，给出一个源于三个二次域的射线类特征标theta级数显式例子，利用Okano方法得到带同余条件的二次型theta级数公式，并证明其类似$\sigma$的系数由分拆生成函数给出。
We generalize a terminating summation formula to a unilateral nonterminating, and further, a bilateral summation formula by a property of analytic functions. The unilateral one is proved to be a $q$-analogue of a $_4F_3$-summation formula. And, an identity unifying Jacobi's triple product identity and the quintuple product identity is obtained as a special case of the bilateral one.
The colored Jones polynomial is a $q$-polynomial invariant of links colored by irreducible representations of a simple Lie algebra. A $q$-series called a tail is obtained as the limit of the $\mathfrak{sl}_2$ colored Jones polynomials $\{J_n(K;q)\}_n$ for some link $K$, for example, an alternating link. For the $\mathfrak{sl}_3$ colored Jones polynomials, the existence of a tail is unknown. We give two explicit formulas of the tail of the $\mathfrak{sl}_3$ colored Jones polynomials colored by $(n,0)$ for the $(2,2m)$-torus link. These two expressions of the tail provide an identity of $q$-seri…
A Dilogarithm Series and a Generalization of a Theorem of Bridgeman
本文利用Abel五项关系推导Rogers双对数的双参数级数恒等式，特化后得到含Lucas序列的级数，推广了Bridgeman关于Pell方程解的级数结果。
Franklin's argument proves an identity of Zagier
最近Zagier证明了一个显着的q系列身份。我们表明，这个身份也可以证明修改富兰克林的经典证明欧拉的五边形数定理。
In this note, we present a curious $q$-series identity with applications to certain partitions with bounded part differences.
We use the method of tiling to give elementary combinatorial proofs of some celebrated $q$-series identities, such as Jacobi triple product identity, Rogers-Ramanujan identities, and some identities of Rogers. We give a tiling proof of the q-binomial theorem and a tiling interpretation of the q-binomial coefficients. A new generalized $ k $-product $q$-series identity is also obtained by employing the `tiling-method', wherein the generating function of the set of all possible tilings of a rectangular board is computed in two different ways to obtain the desired $q$-series identity. Several new…
We propose and prove a new polynomial identity that implies Schur's partition theorem. We give combinatorial interpretations of some of our expressions in the spirit of Kurşungöz. We also present some related polynomial and $q$-series identities.
A new Ramanujan-like series for $1/π^2$
我们的主要结果是$1/π^2$的一个新的Ramanujan-like级数的WZ-证明和一个包含三个级数的超几何恒等式。
Theta function identities and q series
该研究从离散傅里叶变换特征向量出发，建立theta函数恒等式，给出经典四阶恒等式、Landen变换与q级数的初等证明，并推导Rogers-Ramanujan型恒等式和theta函数恒等式的关联。
Touchard's identity and a detailed determination of the radius of convergence of the Catalan series
本文通过触德递推关系证明了卡特兰级数收敛半径为1/4，并结合经典下界得出精确结果。
An exact transformation, which we call the \emph{master identity}, is obtained for the first time for the series $\sum_{n=1}^{\infty}σ_{a}(n)e^{-ny}$ for $a\in\mathbb{C}$ and Re$(y)>0$. New modular-type transformations when $a$ is a non-zero even integer are obtained as its special cases. The precise obstruction to modularity is explicitly seen in these transformations. These include a novel companion to Ramanujan's famous formula for $ζ(2m+1)$. The Wigert-Bellman identity arising from the $a=0$ case of the master identity is derived too. When $a$ is an odd integer, the well-known modular tran…
By using some hypergeometric series identities, we prove two supercongruences on truncated hypergeometric series, one of which is related to a modular Calabi--Yau threefold, and the other is regarded as $p$-adic analogue of an identity due to Ramanujan.
A divisor generating q-series identity and its applications to probability theory and random graphs
本文系统研究Uchimura除数生成q-级数恒等式的各类推广，建立统一理论，并将其应用于概率论中堆和随机无环有向图的分析。
Symmetric square type $L$-series
本文构造了基于判别形式Weil表示的向量值模形式的对称平方类型 $L$-函数，并研究其Hecke算子和积分表示的性质。
On a certain metaplectic Eisenstein series and the twisted symmetric square L-function
本文在Bump和Ginzburg的基础上，利用GL（r）的二重覆盖上的Eisenstein级数，得到了GL（r）的扭对称平方L函数的整表示.利用这一点，我们证明了GL（r）的（不完全）扭对称平方L函数对Re（s）> 1是全纯的.在本文中，我们将更精确地确定这个Eisenstein级数的可能极点，并证明了（不完全）扭对称平方L函数除了在s = 0和s = 1时可能的单极点之外是完整的.
Theta Series of Unimodular Lattices, Combinatorial Identities and Weighted Symmetric Polynomials
我们找到了两个组合恒等式的根格的θ系列的有限维单李代数的$D_{4n}$型和陪集在其积分型，在著名的Essenstein系列$E_4(z)$和Ramanujan系列$\Dlt_{24}(z)$。利用这两个恒等式，我们确定了某些正定偶幺模格的无限族的θ级数，这些无穷族是由$D_{2n}$型有限维单李代数的根格的有限个拷贝胶合而成的.证明了这些θ级数是$E_4(z)$和$\Dlt_{24}(z)$两个固定多项式族的加权对称多项式。
On recursions for coefficients of mock theta functions
我们使用一个广义的Lambert级数恒等式，由于第一作者提出的q-级数证明最近的结果Imamoglu，Raum和Richter有关的递归公式的两个三阶模拟θ函数的系数。此外，我们讨论了这个身份的其他模拟θ函数的应用。
A remarkable basic hypergeometric identity
本文给出当基q取单位根时截断基本超几何级数之商的闭形式，贡献了一个新的基本超几何恒等式。
Six proofs for an identity of the Lah numbers
本文分别利用归纳法、Lah数的母函数、Chu-Vandermonde求和公式、求反演公式、Gauss超几何级数和第一类Stirling数的两个母函数，对Lah数的一个恒等式给出了6个证明.
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。