David M. Bradley
We define a class of expressions for the multiple zeta function, and show how to determine whether an expression in the class vanishes identically. The class of such identities, which we call partition identities, is shown to coincide with the class of identities that can be derived as a consequence of the stuffle multiplication rule for multiple zeta values.
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In this paper, we prove a new identity for values of the Hurwitz zeta function which contains as particular cases Koecher's identity for odd zeta values, the Bailey-Borwein-Bradley identity for even zeta values and many other interesting formulas related to values of the Hurwitz zeta function. We also get an extension of the bivariate identity of Cohen to values of the Hurwitz zeta function. The main tool we use here is a construction of new Markov-WZ pairs. As application of our results, we prove several conjectures on supercongruences proposed by J. Guillera, W. Zudilin, and Z.-W. Sun.
We use visible point vector identities to examine polylogarithms in the neighbourhood of the Riemann zeta function zeroes. New formulas limiting to the trivial zeroes and to the critical line on the zeta function are given. Similar results arise from Euler-Zagier sums given in Bailey, Borwein and Crandall [4] providing new infinite product identities.
We exploit transformations relating generalized $q$-series, infinite products, sums over integer partitions, and continued fractions, to find partition-theoretic formulas to compute the values of constants such as $π$, and to connect sums over partitions to the Riemann zeta function, multiple zeta values, and other number-theoretic objects.
We examine "partition zeta functions" analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties -- those summed over partitions of fixed length -- which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahon's partial fraction decomposition of the generating function for partitions…
We introduce and survey results on two families of zeta functions connected to the multiplicative and additive theories of integer partitions. In the case of the multiplicative theory, we provide specialization formulas and results on the analytic continuations of these "partition zeta functions", find unusual formulas for the Riemann zeta function, prove identities for multiple zeta values, and see that some of the formulas allow for $p$-adic interpolation. The second family we study was anticipated by Manin and makes use of modular forms, functions which are intimately related to integer par…
Partitions, Multiple Zeta Values and the q-bracket
本文建立分拆多项式函数空间与多重 zeta 值的 q-模拟之间的框架，刻画 q→1 极限并双向推导关系，还得到新的拟模 q 级数例子。
We find two involutions on partitions that lead to partition identities for Ramanujan's third order mock theta functions $ϕ(-q)$ and $ψ(-q)$. We also give an involution for Fine's partition identity on the mock theta function f(q). The two classical identities of Ramanujan on third order mock theta functions are consequences of these partition identities. Our combinatorial constructions also apply to Andrews' generalizations of Ramanujan's identities.
Identities involving partitions with distinct even parts and $4$-regular partitions
本文在已知不同偶数部分分拆数与4-正则分拆数相等的基础上，证明了一类受限的不同偶数部分分拆与受限4-正则分拆之间的恒等式。
We show that, in many cases, there are infinitely many sets of partitions corresponding to a single analytical Rogers-Ramanujan type identity. This means that a single analytical Rogers-Ramanujan type identity implies the existence of bijections among infinitely many sets of partitions. We also give an explicit description of these infinite sets coming from the sum side of the analytical identity explaining how to interpret the sum side combinatorially as the generating function of the partitions considered. Moreover, we give a new infinite familiy of Rogers-Ramanujan type identities obtained …
Combinatorial proofs of several partition identities of Andrews and Merca
本文基于Fu和Tang的双射，为Andrews和Merca的若干分拆恒等式提供了组合证明，并引入两种分拆权重扩展了其中一个恒等式。
Higher level $q$-multiple zeta values with applications to quasimodular forms and partitions
本文统一了高阶$q$-多重zeta值与拟模形式及分拆理论，证明任意权重$k\geq2$、水平$N$的拟模形式可由水平$N$的$q$-多重zeta值表示，并构造了整数系数情形的完全可加生成集，给出多种数论计算示例。
Using a new graphical representation for partitions, the author obtains a family of partition identities associated with partitions into distinct parts of an arithmetic progression, or, more generally, with partitions into distinct parts of a set that is a finite union of arithmetic progressions associated with a modular sum-free Sidon set. Partition identities are also constructed for sets associated with modular sum-free sets.
Using the slow triangle map (a type of multi-dimensional continued fraction algorithm), we exhibit a method for generating any number of new identities for subsets of integer partitions.
We prove a new polynomial refinement of the Capparelli's identities. Using a special case of Bailey's lemma we prove many infinite families of sum-product identities that root from our finite analogues of Capparelli's identities. We also discuss the $q\mapsto 1/q$ duality transformation of the base identities and some related partition theoretic relations.
In this note we introduce multi-interpolated multiple zeta values. We provide a basic decomposition of these objects involving ordered partitions. We also obtain identities for special instances of multi-interpolated multiple zeta values $ζ^{\vec{t}}(\{s\}_k)$, generalizing earlier results. Moreover, we introduce a product for multi-interpolated multiple zeta values.
The celebrated Rogers-Ramanujan identities equate the number of integer partitions of $n$ ($n\in\mathbb N_0$) with parts congruent to $\pm 1 \pmod{5}$ (respectively $\pm 2 \pmod{5}$) and the number of partitions of $n$ with super-distinct parts (respectively super-distinct parts greater than $1$). In this paper, we establish companion identities to the Rogers-Ramanujan identities on the number of parts in all partitions of $n$ of the aforementioned types, in the spirit of earlier work by Andrews and Beck on a partition identity of Euler.
The number of partitions of n into parts divisible by a or b equals the number of partitions of n in which each part and each difference of two parts is expressible as a non-negative integer combination of a or b. This generalizes identities of MacMahon and Andrews. The analogous identities for three or more integers (in place of a,b) hold in certain cases.
We propose and recursively prove polynomial identities which imply Capparelli's partition theorems. We also find perfect companions to the results of Andrews, and Alladi, Andrews and Gordon involving $q$-trinomial coefficients. We follow Kurşungöz's ideas to provide direct combinatorial interpretations of some of our expressions. We use of the trinomial analogue of Bailey's lemma to derive new identities. These identities relate triple sums and products. A couple of new Slater type identities are also noted.
Linear identities for partition pairs with $5$-cores
本文通过theta函数恒等式推导出5核分区对数A₅(n)的线性恒等式及同余式。
In this note we conjecture Rogers-Ramanujan type colored partition identities for an array with odd number of rows w such that the first and the last row consist of even positive integers. In a strange way this is different from the partition identities for the array with odd number of rows w such that the first and the last row consist of odd positive integers -- the partition identities conjectured by S. Capparelli, A. Meurman, A. Primc and the author and related to standard representations of the affine Lie algebra of type $C^{(1)}_\ell$ for $w=2\ell+1$. The conjecture is based on numerical…
On a pair of three-colored (mod 10) partition identities
本文证明了一对模10的三色分拆恒等式，将三色不同部分分拆的生成函数与Rogers-Ramanujan乘积联系起来，并利用Maple验证函数方程。
Multiple zeta values, WZ-pairs and infinite sums computations
本研究结合Wilf-Zeilberger对方法与多重zeta值系统理论，证明了Z.W. Sun提出的大量级数恒等式，破解了诸多长期悬而未决的猜想。
SIP classes and four-parameter partition identities
本文扩展了SIP类，提出统一方法研究四参数划分权重，并给出了与该类划分相关的恒等式及生成函数。
Recently Corteel and Welsh outlined a technique for finding new sum-product identities by using functional relations between generating functions for cylindric partitions and a theorem of Borodin. Here, we extend this framework to include very general product-sides coming from work of Han and Xiong. In doing so, we are led to consider structures such as weighted cylindric partitions, symmetric cylindric partitions and weighted skew double shifted plane partitions. We prove some new identities and obtain new proofs of known identities, including the Göllnitz-Gordon and Little Göllnitz identitie…
Signed Partitions and Rogers-Ramanujan type Identities
本文研究 Rogers–Ramanujan 型恒等式的带符号分拆解释，提出不同于 Andrews 的 Göllnitz–Gordon 恒等式新解释，并给出解析与双射证明。
In a work of 1995, Alladi, Andrews, and Gordon provided a generalization of the two Capparelli identities involving certain classes of integer partitions. Inspired by that contribution, in particular as regards the general setting and the tools the authors employed, we obtain new partition identities by identifying further sets of partitions that can be explicitly put into a one-to-one correspondence by the method described in the 1995 paper. As a further result, although of a different nature, we obtain an analytical identity of Rogers-Ramanujan type, involving generating functions, for a cla…
We utilize false theta function results of Nathan Fine to discover three new partition identities involving weights. These relations connect Göllnitz--Gordon type partitions and partitions with distinct odd parts, partitions into distinct parts and ordinary partitions, and partitions with distinct odd parts where the smallest positive integer that is not a part of the partition is odd and ordinary partitions subject to some initial conditions, respectively. Some of our weights involve new partition statistics, one is defined as the number of different odd parts of a partition larger than or eq…
Let S and T be sets of positive integers and let a be a fixed positive integer. An a-shifted partition identity has the form p(S,n)=p(T,n-a), for all n greater or equal to a. Here p(S,n) is the number partitions of n whose parts are elements of S. For all known nontrivial shifted partition identities, the sets S and T are unions of arithmetic progressions modulo M for some M. In 1987, Andrews found two 1-shifted examples (M=32, 40) and asked whether there were any more. In 1989, Kalvade responded with a further six. In 2000, the first author found 59 new 1-shifted identities using a computer s…
We prove seven of the Rogers-Ramanujan type identities modulo $12$ that were conjectured by Kanade and Russell. Included among these seven are the two original modulo $12$ identities, in which the products have asymmetric congruence conditions, as well as the three symmetric identities related to the principally specialized characters of certain level $2$ modules of $A_9^{(2)}$. We also give reductions of four other conjectures in terms of single-sum basic hypergeometric series.
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…
Linear identities for partition pairs with 4-cores
利用基本q-级数技巧和3-分拆公式，确定了具有4-核的n的分拆对个数A_4(n)的无穷线性恒等式族，并推导出相应的同余式。
Combinatorial interpretation of the Schlesinger-Zudilin stuffle product
本文通过将Schlesinger-Zudilin多重q-泽塔值解释为特殊标记分块的生成级数，推导其拟混洗积的显式公式，结合对偶性描述及Bachmann猜想，完成了多重q-泽塔值猜想结构的标记分块描述。
This paper is about the determinantal identities associated with the Ihara (Ih) zeta function of a non directed graph and the Bowen-Lanford (BL) zeta function of a directed graph. They will be called the Ih and the BL identities in this paper. We show that the Witt identity (WI) is a special case of the BL identity and inspired by the links the WI has with Lie algebras and combinatorics we investigate similar aspects of the Ih and BL identities. We show that they satisfy generalizations of the Strehl identity and Carlitz, Metropolis-Rota relations and each one of them can be interpreted as the…
Identities involving the number of missing integers in partitions - combinatorial proofs
针对Bhoria等人关于分拆中缺失整数个数的相关结果，本文提供了组合证明，丰富了分拆组合学领域的研究内容。
Kanade and Russell conjectured several Rogers-Ramanujan-type partition identities, some of which are related to level $2$ characters of the affine Lie algebra $A_9^{(2)}$. Many of these conjectures have been proved by Bringmann, Jennings-Shaffer and Mahlburg. We give new proofs of five conjectures first proved by those authors, as well as four others that have been open until now. Our proofs for the new cases use quadratic transformations for Askey-Wilson and Rogers polynomials. We also obtain some related results, including a new proof of a partition identity conjectured by Capparelli and fir…
Multiple Rogers--Ramanujan type identities for torus links
本文为 $(2,2k)$ 环面链环的 Quot 和 motivic Cohen--Lenstra zeta 函数建立 $k$ 重求和表达式，导出多重 Rogers--Ramanujan 型恒等式并证实 Huang--Jiang 猜想。
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.
In this paper we present a new class of integer partition identities. The number of partitions with d-distant parts can be represented as a sum of the number of partitions with 1-distant parts whose even parts are greater than twice the number of odd parts. We also provide a direct bijection between these classes of partitions.
In our previous paper, we determined a unified combinatorial framework to look at a large number of colored partition identities, and studied the five identities corresponding to the exceptional modular equations of prime degree of the Schroeter, Russell and Ramanujan type. The goal of this paper is to use the master bijection of our previous work to show combinatorially several new and highly nontrivial colored partition identities. We conclude by listing a number of further interesting identities of the same type as conjectures.
We construct a family of partition identities which contain the following identities: Rogers-Ramanujan-Gordon identities, Bressoud's even moduli generalization of them, and their counterparts for overpartitions due to Lovejoy et al. and Chen et al. We obtain unusual companion identities to known theorems as well as to the new ones in the process. The proof is, against tradition, constructive and open to automation.
Combinatorial proof of identities involving partitions with distinct even parts and 4-regular partitions
针对Andrews与Bachraoui用基本超几何级数证明并寻求组合解释的互异偶数部分分拆与4-正则分拆恒等式，本文通过构造双射给出组合证明。
Recently, Merca and Yee proved some partition identities involving two new partition statistics. We refine these statistics and generalize the results of Merca and Yee. We also correct a small mistake in a result of Merca and Yee.
We show that, up to multiplication by a factor $\frac{1}{(cq;q)_{\infty}}$, the weighted words version of Capparelli's identity is a particular case of the weighted words version of Primc's identity. We prove this first using recurrences, and then bijectively. We also give finite versions of both identities.
We prove two partition identities which are dual to the Rogers-Ramanujan identities. These identities are inspired by (and proved using) a correspondence between three kinds of objects: a new type of partitions (neighborly partitions), monomial ideals and some infinite graphs.
Identities involving partitions with distinct odd parts and no parts congruent to 2 mod 4
本文针对Mahanta和Saikika提出的恒等式给出了q级数证明，并利用不同奇数部分分拆与无部分同余于2 mod 4分拆数量相等的性质，给出了相关恒等式的组合证明。
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…
We give a series of recursive identities for the number of partitions with exactly $k$ parts and with constraints on both the minimal difference among the parts and the minimal part. Using these results we demonstrate that the number of partitions of $n$ is equal to the number of partitions of $2n+d{n \choose 2}$ with $n$ $d$-distant parts. We also provide a direct proof for this identity. This work is the result of our aim at finding a bijective proof for Rogers-Ramanujan identities.
We provide a bijective map from the partitions enumerated by the series side of the Rogers-Selberg mod 7 identities onto partitions associated with a special case of Basil Gordon's combinatorial generalization of the Rogers-Ramanujan identities. The implications of applying the same map to a special case of David Bressoud's even modulus analog of Gordon's theorem are also explored.
Neighborly partitions, hypergraphs and Gordon's identities
本文证明了一族与安德鲁斯-戈登恒等式对偶的划分恒等式，其灵感来自特殊划分与超图的对应关系，证明方法采用组合交换代数。
We give relations between the joint distributions of multiple hook lengths and of frequencies and part sizes in partitions, extending prior work in this area. These results are discovered by investigating truncations of the Han/Nekrasov-Okounkov hooklength formula and of (k,j)-colored partitions, a unification of k-colored partitions and overpartitions. We establish the observed relations at the constant and linear terms for all n, and for j=2 in their quadratic term, with the associated hook/frequency identities. Further results of this type seem likely.
The combinatorics of some two-color partition identities
本研究针对Andrews、EI Bachraoui及Chen和Zhou提出的双色分拆恒等式相关问题，给出了已有恒等式的组合证明，还推导了若干新的双色分拆恒等式并提供组合证明，推进了分拆组合学领域的相关研究。
We generalise Euler's partition theorem involving odd parts and different parts for all moduli and provide new companions to Rogers-Ramanujan- Andrews-Gordon identities related to this theorem.
Combinatorial perspectives on identities for partitions with distinct even parts
本文从组合学角度研究具有不同偶数部分的划分，通过与带符号划分和双色划分的联系，获得新的划分身份，并通过双射证明解决相关组合问题。
We prove a theorem which add a new member to Rogers-Ramanujan identities. This new member counts partitions with different type of constraints on even and odd parts. Generalizing this theorem, we obtain two family of partition identities of Rogers-Ramanujan type.
We generalize the theory of linked partition ideals due to Andrews using finite automata in formal language theory and apply it to prove three Rogers--Ramanujan type identities of modulo 14 that were posed by Nandi through vertex operator theoretic construction of the level 4 standard modules of the affine Lie algebra $A^{(2)}_{2}$.
Based on weighted noncrossing partitions of type B, we obtain type B analogues of Coker's identities on the Narayana polynomials. A parity reversing involution is given for the alternating sum of Narayana numbers of type B. Moreover, we find type B analogues of the refinements of Coker's identities due to Chen, Deutsch and Elizalde. By combinatorial constructions, we provide type B analogues of three identities of Mansour and Sun also on the Narayana polynomials.
Elementary proofs are given for sums of Schur functions over partitions into at most n parts each less than or equal to m for which i) all parts are even, ii) all parts of the conjugate partition are even. Also, an elementary proof of a recent result of Ishikawa and Wakayama is given.
In this paper, we give a conjecture, which generalises Euler's partition theorem involving odd parts and different parts for all moduli. We prove this conjecture for two family partitions. We give $q$-difference equations for the related generating function if the moduli is three. We provide new companions to Rogers-Ramanujan-Andrews-Gordon identities under this conjecture.
In these two companion papers, we give infinite families of partition identities which generalise Primc's and Capparelli's identities, and study their consequences on the theory of crystal bases of the affine Lie algebra $A_{n-1}^{(1)}.$ In this first paper, we focus on combinatorial aspects. We give a $n^2$-coloured generalisation of Primc's identity by constructing a $n^2 \times n^2$ matrix of difference conditions, Primc's original identities corresponding to $n=2$ and $n=3$. While most coloured partition identities in the literature connect partitions with difference conditions to partitio…
Signed Counting on Restricted Partitions and Combinatorial Proofs of Three Identities
该研究受相关组合方法启发，对\(l\)-正则划分及其变体的三个恒等式给出组合证明，其中两个恒等式此前由他人通过解析方法建立，此为恒等式证明提供了新的组合证明方式。
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