Johann Cigler
We give simple elementary proofs of Bressoud's and Schur's polynomial versions of the Rogers-Ramanujan identities
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A new recursion in only one variable allows very simple verifications of Bressoud's polynomial identities, which lead to the Rogers-Ramanujan identities. This approach might be compared with an earlier approach due to Chapman. Applying the $q$-Chu-Vandermonde convolution, as suggested by Cigler, makes the computations particularly simple and elementary. The same treatment is also applied to the Santos polynomials and perhaps more polynomials from a list of Rogers-Ramanujan like polynomials \cite{Sills}.
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…
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.
The Rogers-Ramanujan Identities and Cauchy's Identity
本文利用Schur函数的Cauchy恒等式对Rogers-Ramanujan恒等式进行了研究。
Refinements of the classical Rogers-Ramanujan identities are given in which some parts are weighted. Combinatorial interpretations refining MacMahon's results are corollaries.
We prove two new summation formulae of Hall-Littlewood polynomials over partitions into bounded parts and derive some new multiple $q$-identities of Rogers-Ramanujan type.
On Bilateral Multiple Sums and Rogers-Ramanujan Type Identities
本文通过双侧双和罗杰斯-拉马努金恒等式建立新恒等式，并利用基本超几何级数理论和积分方法证明，推导出多种新的多重和罗杰斯-拉马努金型恒等式。
By studying non-commutative series in an infinite alphabet we introduce shift-plethystic trees and a class of integer compositions as new combinatorial models for the Rogers-Ramanujan identities. We prove that the language associated to shift-plethystic trees can be expressed as a non-commutative generalization of the Rogers-Ramanujan continued fraction. By specializing the noncommutative series to $q$-series we obtain new combinatorial interpretations to the Rogers-Ramanujan identities in terms of signed integer compositions. We introduce the operation of shift-plethysm on non-commutative ser…
Resorting to the recursions satisfied by the polynomials which converge to the right hand sides of the Rogers-Ramanujan type identities given by Sills and a determinant method presented in a paper by Ismail-Prodinger-Stanton, we obtain many new one-parameter generalizations of the Rogers-Ramanujan type identities, such as a generalization of the analytic versions of the first and second Göllnitz-Gordon partition identities, and generalizations of the first, second, and third Rogers-Selberg identities.
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 note we show how to rederive the $A_2$ Rogers-Ramanujan identities proven by Andrews, Schilling and Warnaar using cylindric partitions. This paper is dedicated to George Andrews for his $80^{th}$ birthday.
In a recent work, Andrews defined the singular overpartitions with the goal of presenting an overpartition analogue to the theorems of Rogers--Ramanujan type for ordinary partitions with restricted successive ranks. As a small part of his work, Andrews noted two congruences modulo $3$ for the number of singular overpartitions prescribed by parameters $k=3$ and $i=1$. It should be noticed that this number equals the number of the Rogers--Ramanujan--Gordon type overpartitions with $k=i=3$ which come from the overpartition analogue of Gordon's Rogers--Ramanujan partition theorem introduced by Che…
In 2010, Andrews considers a variety of parity questions connected to classical partition identities of Euler, Rogers, Ramanujan and Gordon. As a large part in his paper, Andrews considered the partitions by restricting the parity of occurrences of even numbers or odd numbers in the Rogers-Ramanujan-Gordon type. The Rogers-Ramanujan-Gordon type partition was defined by Gordon in 1961 as a combinatorial generalization of the Rogers-Ramaujan identities with odd moduli. In 1974, Andrews derived an identity which can be considered as the generating function counterpart of the Rogers-Ramanujan-Gord…
We give a combinatorial proof of the first Rogers-Ramanujan identity by using two symmetries of a new generalization of Dyson's rank. These symmetries are established by direct bijections.
Some $q$-transformation formulas and Rogers-Ramanujan type identities
本文利用 Liu 变换公式结合其他 $q$-级数求和，推导参数化恒等式，并从中获得 75 个已知及若干新的 Rogers-Ramanujan 型恒等式。
Rogers-Ramanujan Type $q$-Exponential Operator and Stieljes-Widgert Polynomials
研究运用Rogers-Ramanujan型$q$-指数算子，建立非规范化齐次Stieljes-Wigert多项式的生成函数及Mehler、Rogers公式。
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.
We extend partition-theoretic work of Andrews, Bressoud, and Burge to overpartitions, defining the notions of successive ranks, generalized Durfee squares, and generalized lattice paths, and then relating these to overpartitions defined by multiplicity conditions on the parts. This leads to many new partition and overpartition identities, and provides a unification of a number of well-known identities of the Rogers-Ramanujan type. Among these are Gordon's generalization of the Rogers-Ramanujan identities, Andrews' generalization of the Göllnitz-Gordon identities, and Lovejoy's ``Gordon's theor…
For $k\geq i\geq 1$, let $B_{k,i}(n)$ denote the number of partitions of $n$ such that part 1 appears at most $i-1$ times, two consecutive integers l and $l+1$ appear at most $k-1$ times and if l and $l+1$ appear exactly $k-1$ times then the total sum of the parts l and $l+1$ is congruent to $i-1$ modulo 2. Let $A_{k,i}(n)$ denote the number of partitions with parts not congruent to $i$, $2k-i$ and $2k$ modulo $2k$. Bressoud's theorem states that $A_{k,i}(n)=B_{k,i}(n)$. Corteel, Lovejoy, and Mallet found an overpartition analogue of Bressoud's theorem for $i=1$, that is, for partitions not co…
A New Series for Rogers-Ramanujan-Gordon Identities when k = 3
本文为k=3的Rogers-Ramanujan-Gordon恒等式引入新级数，用基础分拆及前后向移动给出组合解释，并展望推广方向。
We present a generalization, which we call (k,m)-rank, of Dyson's notion of rank to integer partitions with k successive Durfee rectangles and give two combinatorial symmetries associated with this new definition. We prove these symmetries bijectively. Using the two symmetries we give a new combinatorial proof of generalized Roger-Ramanujan identities. We also describe the relationship between (k,m)-rank and Garvan's k-rank.
A generalized Bailey pair, which contains several special cases considered by Bailey (\emph{Proc. London Math. Soc. (2)}, 50 (1949), 421--435), is derived and used to find a number of new Rogers-Ramanujan type identities. Consideration of associated $q$-difference equations points to a connection with a mild extension of Gordon's combinatorial generalization of the Rogers-Ramanujan identities (\emph{Amer. J. Math.}, 83 (1961), 393--399). This, in turn, allows the formulation of natural combinatorial interpretations of many of the identities in Slater's list (\emph{Proc. London Math. Soc. (2)} …
We study the generating functions for cylindric partitions with profile $(c_1,c_2,c_3)$ for all $c_1,c_2,c_3$ such that $c_1+c_2+c_3=5$. This allows us to discover and prove seven new $A_2$ Rogers-Ramanujan identities modulo $8$ with quadruple sums, related with work of Andrews, Schilling, and Warnaar.
We present two general finite extensions for each of the two Rogers-Ramanujan identities. Of these one can be derived directly from Watson's transformation formula by specialization or through Bailey's method, the second similar formula can be proved either by using the first formula and the q-Gosper algorithm, or through the so-called Bailey lattice.
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.
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.
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…
"Overpartitionized" Rogers--Ramanujan type identities
本文通过对经典 $q$-级数进行代数操作，为 Rogers--Ramanujan 等恒等式提供了基于过度划分的组合解释，并给出了双射证明。
Lattice paths and the Rogers--Ramanujan--Gordon type theorems with parity considerations
本文通过三类单位步格路径与 Rogers–Ramanujan–Gordon 型分拆之间的双射，将 Andrews 的奇偶性分拆定理对应到格路径，并获得新结论。
Double-sum Rogers-Ramanujan type identities
本文通过Chebyshev多项式展开式与正交关系，推导出双和罗格斯-拉曼努金型恒等式，扩展了安德鲁斯、史、孙和姚等人的已有结果。
Sequences of odd length in strict partitions IV: the combinatorics of parameterized Rogers-Ramanujan type identities
本文延续严格分拆中奇长度序列的研究，用双射方法重建并参数化若干 Rogers-Ramanujan 型恒等式，同时推广 Chen-Yin 的相关结果。
Bilateral Bailey pairs and Rogers-Ramanujan type identities
本文利用q-二项式定理发现双边Bailey对，结合双边Bailey引理、链和格，统一推导出多种Rogers-Ramanujan型恒等式及Appell-Lerch、Andrews-Gordon和Hecke型级数恒等式。
A general $q$-series transformation and its applications to multi-sum Rogers-Ramanujan-Slater identities
本文提出一个包含 Nahm 级数变换的一般 $q$-级数变换，并由此导出 ${}_{r+1}ϕ_r$ 级数的新变换公式及多和 Rogers-Ramanujan-Slater 型 $q$-恒等式。
Two new Bailey pairs and their $q$-identities of Rogers-Ramanujan type modulo 15, 24, and 30
该研究通过推广 Euler 五边形数定理构造两个新 Bailey 对，并结合 Bailey 引理得到模 15、24、30 的 Rogers–Ramanujan 型 $q$-恒等式与级数变换。
We show that a strange group theoretical formula of Hall follows from the Rogers-Ramanujan identities.
Bilateral identities of the Rogers--Ramanujan type
本文利用解析方法建立一系列 Rogers–Ramanujan 型双边及多边恒等式，推广多项经典恒等式，并给出闭式求和。
Polynomial generalizations of all 130 of the identities in Slater's list of identities of the Rogers-Ramanujan type are presented. Furthermore, duality relationships among many of the identities are derived. Some of the these polynomial identities were previously known but many are new. The author has implemented much of the finitization process in a Maple package which is available for free download from the author's website.
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.
Some multidimensional Rogers--Ramanujan type identities
本文通过围道积分法建立了双重级数到单重级数的带参数化简公式，推广了多项已有结果并发现了三重和推广，最终给出了若干带参数或不带参数的多维 Rogers-Ramanujan 类型恒等式。
A new type of polynomial analogue of the Rogers-Ramanujan identities is proven. Here the product-side of the Rogers-Ramanujan identities is replaced by a partial theta sum and the sum-side by a weighted sum over Schur polynomials.
Lattice paths and Rogers--Ramanujan--Gordon type overpartitions
本文建立了 Rogers-Ramanujan-Gordon 型超划分与四类酉步长格点路径的双射关系，将 Chen、Sang 和 Shi 的定理转化为格点路径形式，并基于奇偶性约束推导出超划分的新结果。
Several new product identities in relation to two-variable Rogers-Ramanujan type sums and mock theta functions
本研究将双变量乘积恒等式的级数展开视为拟周期亚纯函数向量空间的典范基表示，发现了多组新的乘积与Rogers-Ramanujan型和恒等式，揭示了乘积恒等式与Rogers-Ramanujan恒等式间的对应关系并给出应用。
The Bailey lemma is a famous tool to prove Rogers-Ramanujan type identities. We use shifted versions of the Bailey lemma to derive $m$-versions of multisum Rogers-Ramanujan type identities. We also apply this method to the Well-Poised Bailey lemma and obtain a new extension of the Rogers-Ramanujan identities.
Asymmetric bilateral Bailey pairs and Rogers-Ramanujan type identities
本文通过非对称双侧Bailey对推导出罗杰斯-拉马努金型、安德鲁斯-戈登型及假θ函数恒等式，并基于Dousse等人构建的Bailey晶格得到非对称双侧Bailey引理，进一步推导出Appell-Lerch级数恒等式。
We find involutions for three Rogers-Ramanujan-Gordon type identities obtained by Andrews on the generating functions for partitions with part difference and parity restrictions.
In this paper, we investigate several infinite products with vanishing Taylor coefficients in arihmetic progressions. These infinite products are closely related to the Rogers--Ramanujan continued fraction. Moreover, a handful of new identities involving Ramanujan's parameters will be established.
A New (But Very Nearly Old) Proof of the Rogers-Ramanujan Identities
本文利用Rogers 1894年论文中已有的要素，为Rogers-Ramanujan恒等式提供了一个比原证明更简单的新证明。
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.
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 …
A proof of $J$-generalization of the Rogers-Ramanujan-Gordon identities via commutative algebra
本文通过交换代数方法证明了Coulson等人提出的更广泛恒等式族，其中包含罗杰斯-拉马努金-戈登恒等式，并将相关生成函数与分层代数的Hilbert-Poincaré级数联系起来。
We give an easy approach to L. Slater's Bailey pairs A(1)-A(8) with the help of q-Lucas polynomials.
We prove an identity for Hall--Littlewood symmetric functions labelled by the Lie algebra A_2. Through specialization this yields a simple proof of the A_2 Rogers--Ramanujan identities of Andrews, Schilling and the author.
Sequences of odd length in strict partitions I: the combinatorics of double sum Rogers-Ramanujan type identities
本文将严格分拆的三元生成函数表为双重和 $q$-级数，据此为近期多个双重和 Rogers-Ramanujan 型恒等式提供分拆理论解释及 Franklin 型对合证明。
New proofs of the septic Rogers-Ramanujan identities
本文为传统上与模7、14和28相关的十二个Rogers-Ramanujan型恒等式提供了新证明。
In this we paper we prove several new identities of the Rogers-Ramanujan-Slater type. These identities were found as the result of computer searches. The proofs involve a variety of techniques, including series-series identities, Bailey pairs, a theorem of Watson on basic hypergeometric series, generating functions and miscellaneous methods.
Basil Gordon, in the sixties, and George Andrews, in the seventies, generalized the Rogers-Ramanujan identities to higher moduli. These identities arise in many areas of mathematics and mathematical physics. One of these areas is representation theory of infinite dimensional Lie algebras, where various known interpretations of these identities have led to interesting applications. Motivated by their connections with Lie algebra representation theory, we give a new interpretation of a sum related to generalized Rogers-Ramanujan identities in terms of multi-color partitions.
We give another proof of the second Rogers-Ramanujan identity by Kashiwara crystals.
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.
We refine and generalise a Rogers-Ramanujan type partition identity arising from crystal base theory. Our proof uses the variant of the method of weighted words recently introduced by the first author.
We report on findings of a variant of ${\texttt{IdentityFinder}}$ - a Maple program that was used by two of the authors to conjecture several new identities of Rogers-Ramanujan kind. In the present search, we modify the parametrization of the search space by taking into consideration several aspects of Lepowsky and Wilson's $Z$-algebraic mechanism and its variant by Meurman and Primc. We search for identities based on forbidding the appearance of "flat" partitions as sub-partitions. Several new identities of Rogers-Ramanujan-MacMahon type are found and proved.
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