Nayandeep Deka Baruah, Nilufar Mana Begum, Hirakjyoti Das
Relations involving the Rogers-Ramanujan continued fractions $R(q),$ $R(q^3 ),$ and $R(q^4)$ are used to find new generating functions and congruences modulo 5 and 25 for 3-core, 4-core, 4-regular, and colored partition functions.
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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。
Let $R(q)$ be the Rogers-Ramanujan continued fraction. We give different proofs of two complementary relations for $R(q)$ given by Ramanujan and proved by Watson and Ramanathan. Our proofs only use product expansions for classical Jacobi theta functions.
It is shown that (two-variable generalizations of) more than half of Slater's list of 130 Rogers-Ramanujan identities (L. J. Slater, Further identities of the Rogers-Ramanujan type, \emph{Proc. London Math Soc. (2)} \textbf{54} (1952), 147--167) can be easily derived using just three multiparameter Bailey pairs and their associated $q$-difference equations. As a bonus, new Rogers-Ramanujan type identities are found along with natural combinatorial interpretations for many of these identities.
We describe three computer searches (in PARI/GP, Maple, and Mathematica, respectively) which led to the discovery of a number of identities of Rogers-Ramanujan type and identities of false theta functions.
The Sylvester Theorem and the Rogers-Ramanujan Identities over Totally Real Number Fields
本文将西尔维斯特定理与罗杰斯-拉马努金恒等式推广到全实代数数域上的全正代数整数分拆，并进一步给出广义罗杰斯-拉马努金恒等式的另一版本。
The aim of this paper is to study the convergence and divergence of the Rogers-Ramanujan and the generalized Rogers-Ramanujan continued fractions on the unit circle. We provide an example of an uncountable set of measure zero on which the Rogers-Ramanujan continued fraction $R(x)$ diverges and which enlarges a set previously found by Bowman and Mc Laughlin. We further study the generalized Rogers-Ramanujan continued fractions $R_a(x)$ for roots of unity $a$ and give explicit convergence and divergence conditions. As such, we extend some work of Huang towards a question originally investigated …
In a handwritten manuscript published with his lost notebook, Ramanujan stated without proofs forty identities for the Rogers-Ramanujan functions. We observe that the functions that appear in Ramanujan's identities can be obtained from a Hecke action on a certain family of eta products. We establish further Hecke-type relations for these functions involving binary quadratic forms. Our observations enable us to find new identities for the Rogers-Ramanujan functions and also to use such identities in turn to find identities involving binary quadratic forms.
Some New Congruences and Partition-Theoretic Interpretations for the Coefficients of Some Rogers-Ramanujan Type Identities
本文利用正整数的 overpartition 和 colour partition，为某些 Rogers-Ramanujan 型恒等式系数提供了分拆理论解释，并证明了模 2 的幂的无限族同余。
In 2015 Choi, Kim, and Lovejoy studied a weighted partition function, $A_1(m)$, which counted subpartitions with a structure related to the Rogers--Ramanujan identities. They conjectured the existence of an infinite class of congruences for $A_1(m)$, modulo powers of 5. We give an explicit form of this conjecture, and prove it for all powers of 5.
Ramanujan recorded four reciprocity formulas for the Roger-Ramanujan continued fraction. Two reciprocity formulas each are also associated with the Ramanujan--Göllnitz--Gordon continued fraction and a level-13 analog of the Roger-Ramanujan continued fraction. We show that all eight reciprocity formulas are related to a pair of quadratic equations. The solution to the first equation generalizes the golden ratio and is used to set the value of a coefficient in the second equation; and the solution to the second equation gives a pair of values for a continued fraction. We relate the coefficients …
We derive two general transformations for certain basic hypergeometric series from the recurrence formulae for the partial numerators and denominators of two $q$-continued fractions previously investigated by the authors. By then specializing certain free parameters in these transformations, and employing various identities of Rogers-Ramanujan type, we derive \emph{$m$-versions} of these identities. Some of the identities thus found are new, and some have been derived previously by other authors, using different methods. By applying certain transformations due to Watson, Heine and Ramanujan, w…
In this work, we start an investigation of asymmetric Rogers--Ramanujan type identities. The first object is the following unexpected relation $$\sum_{n\ge 0} \frac{(-1)^n q^{3\binom{n}{2}+4n}(q;q^3)_n}{(q^9;q^9)_n} = \frac{(q^{4};q^{6})_\infty (q^{12};q^{18})_\infty}{(q^{5};q^{6})_\infty (q^{9};q^{18})_\infty}$$ and its $a$-generalization. We then use this identity as a key ingredient to confirm a recent conjecture of G. E. Andrews and A. K. Uncu.
In [4] and [5], Folsom presents a family of modular units as higher-level analogues of the Rogers-Ramanujan $q$-continued fraction. These units are constructed from analytic solutions to the higher-order $q$-recurrence equations of Selberg. Here, we consider another family of modular units, which are quotients of Hall-Littlewood $q$-series that appear in the generalized Rogers-Ramanujan type identities of [6]. In analogy with the results of Folsom, we provide a formula for the rank of the subgroup these units generate and show that their specializations at the cusp $0$ generate a subgroup of t…
In this survey article, we present an expanded version of Lucy Slater's famous list of identities of the Rogers-Ramanujan type, including identities of similar type, which were discovered after the publication of Slater's papers, and older identities (such as those in Ramanujan's lost notebook) which were not included in Slater's papers. We attempt to supply the earliest known reference for each identity. Also included are identities of false theta functions, along with their relationship to Rogers-Ramanujan type identities. We also describe several ways in which pairs/larger sets of identitie…
In this part we show that the diophantine equation $X^5+Y^5=\varepsilon^5(1-X^5Y^5)$, where $\varepsilon=\frac{-1+\sqrt{5}}{2}$, has solutions in specific abelian extensions of quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ in which $-d \equiv \pm 1$ (mod $5$). The coordinates of these solutions are values of the Rogers-Ramanujan continued fraction $r(τ)$, and are shown to be periodic points of an algebraic function.
Rogers-Ramanujan continued fraction and approximations to $\mathbf{2π}$
本文利用 Rogers–Ramanujan 连分数与 Ramanujan、Rogers 的模方程构造 $2π$ 的高速递归逼近，分别实现正确位数五倍和十一倍增长。
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 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。
Discussion on some conjectures regarding the periodicity of sign patterns of certain infinite products involving the Rogers-Ramanujan Continued Fractions
本文讨论了涉及Rogers-Ramanujan连分数的无穷级数系数符号模式的猜想，通过证明Baruah和Sarma提出的关于$A(5n)$和$B(5n)$符号模式的猜想在$n=0$时不成立，指出了原猜想的局限性。
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。
We give another proof of the second Rogers-Ramanujan identity by Kashiwara crystals.
Applications of the icosahedral equation for the Rogers-Ramanujan continued fraction
本文利用二十面体方程和模关系，通过Wolfram语言的RootApproximant命令，系统发现并证明了Rogers-Ramanujan连分数在特定指数处的根式闭式表达式，包括一个计算上困难的$R(e^{-\pi\sqrt{48/5}})$值。
Logarithmic moments of the Rogers-Ramanujan continued fraction and odd zeta values
该研究探讨Rogers-Ramanujan连分数的尖点展开对其普通与对数Mellin矩的控制作用，证明相关系数的代数性质，确定归一化矩生成函数的极点，并建立反射恒等式等结果。
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.
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.
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 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.
It is known that $q$-orthogonal polynomials play an important role in the field of $q$-series and special functions. During studying Dyson's "favorite" identity of Rogers--Ramanujan type, Andrews pointed out that the classical orthogonal polynomials also have surprising applications in the world of $q$. By inserting Chebyshev polynomials of the third and the fourth kinds into Bailey pairs, Andrews derived a family of Rogers--Ramanujan type identities and also results related to mock theta functions and Hecke--type series. In this paper, by constructing a new Bailey pair involving Chebyshev pol…
The Rogers--Ramanujan continued fraction
本文综述了罗杰斯-拉马努金连续分数的性质、值、恒等式及推广，涵盖其历史背景与相关数学结果。
Explicit formulas are proved for the $5$-torsion points on the Tate normal form $E_5$ of an elliptic curve having $(X,Y)=(0,0)$ as a point of order $5$. These formulas express the coordinates of points in $E_5[5] - \langle(0,0)\rangle$ as products of linear fractional quantities in terms of $5$-th roots of unity and a parameter $u$, where the parameter $b$ which defines the curve $E_5$ is given as $b=(\varepsilon^5 u^5- \varepsilon^{-5})/(u^5+1)$ and $\varepsilon = (-1+\sqrt{5})/2$. If $r(τ)$ is the Rogers-Ramanujan continued fraction and $b=r^5(τ)$, then the coordinates of points of order $5$…
We prove a q-series identity that generalises Macdonald's A_{2n}^{(2)} eta-function identity and the Rogers-Ramanujan identities. We conjecture our result to generalise even further to also include the Andrews-Gordon identities.
On new identities connecting Ramanujan-Göllnitz-Gordon continued fraction and Ramanujan's continued fraction of order four
本文利用q级数和theta函数理论，建立连接拉马努金-戈尔尼茨-戈登连续分数与四阶拉马努金连续分数的新恒等式，并利用拉马努金的_1ψ_1求和公式获得Lambert级数恒等式。
Here we consider the $q$-series coming from the Hall-Littlewood polynomials, \begin{equation*} R_ν(a,b;q)=\sum_{\substack{λ\\[1pt] λ_1\leq a}} q^{c|λ|} P_{2λ}\big(1,q,q^2,\dots;q^{2b+d}\big). \end{equation*} These series were defined by Griffin, Ono, and Warnaar in their work on the framework of the Rogers-Ramanujan identities. We devise a recursive method for computing the coefficients of these series when they arise within the Rogers-Ramanujan framework. Furthermore, we study the congruence properties of certain quotients and products of these series, generalizing the famous Ramanujan congru…
On the Divergence of the Rogers-Ramanujan Continued Fraction on the Unit Circle
设任意无理数$t \in (0,1)$的连分数展开式记为$[0,a_{1}(t),a_{2}(t),...]$，该连分数展开式的第i次收敛记为$c_{i}(t)/d_{i}(t)$。设\[ S=\{t \in (0,1): a_{i+1}(t) \geq ϕ^{d_{i}(t)} \text{infinitely often}\}, \]，其中$ϕ= (\sqrt{5}+1)/2$ .令$Y_{S} =\{\exp(2 πi t): t \in S \}$。证明了若$y \in Y_{S}$成立，则Rogers-Ramanujan连分数R（y）在y处发散. S是一个不可数的零测度集。它还表明，有一个不可数的点集，$G \subset Y_{S}$，使得如果$y \in G$，则R（y）一般不收敛。进一步证明了对于|y| > 1，R（y）一般不收敛;如果y是一个本原的5 m次单位根，某个$m \in \mathbb{N}$，则R（y）一般收敛.
We highlight the role of q-series techniques in proving identities arising from knot theory. In particular, we prove Rogers-Ramanujan type identities for alternating knots as conjectured by Garoufalidis, Le and Zagier.
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。
We use an integral method to establish a number of Rogers-Ramanujan type identities involving double and triple sums. The key step for proving such identities is to find some infinite products whose integrals over suitable contours are still infinite products. The method used here is motivated by Rosengren's proof of the Kanade-Russell identities.
New Weighted Rogers-Ramanujan Partition Theorems and their Implications
本文件有两个目的。首先，通过考虑Göllnitz的一个深定理的重新表述，我们得到了一个新的加权划分恒等式，涉及Rogers-Ramanujan划分，即划分成至少相差两个的部分。其结果包括雅可比著名的θ函数三重积恒等式、西尔维斯特著名的欧拉定理改进以及某些加权分割恒等式。接下来，通过研究分区与连续的行列上的规定的界限，并取代这些加权罗杰斯-Ramanujan分区，我们得到两组新的定理-一组三个定理涉及分区成部分$\not\equiv 0, \pm i$（$mod$ 6），和一组三个定理涉及分区成部分$\not\equiv 0, \pm i$（$mod$ 7），$i=1,2,3$。
Modularity of certain products of the Rogers-Ramanujan continued fraction
本文研究了形如 $r(τ)^ar(2τ)^b$ 的 Rogers-Ramanujan 连分数乘积的模性，证明了在特定条件下有限个此类函数可生成同余子群 $Γ_1(10)$ 上的模函数域，并利用 Lee 和 Park 的方法及 $η$-商性质建立了相关算术性质。
On Ramanujan's cubic continued fraction and explicit evaluations of theta-functions
研究了Ramanujan三次连分数和θ函数的显式求值
Integrals related with Rogers Ramanujan continued fraction and q-products
本文提出了一种计算一类与拉马努扬连分数和q-积有关的积分的方法。为此，我们简要地探讨了一个微分方程，它将这两个函数联系起来
Sign Patterns and Congruences of certain infinite products involving the Rogers-Ramanujan continued fraction
本文研究涉及Rogers-Ramanujan连分数的无穷乘积系数符号规律，证明特定算术级数中系数的正负性，并发现若干模3、模4和模15的同余式。
The famous Rogers-Ramanujan and Andrews--Gordon identities are embedded in a doubly-infinite family of Rogers-Ramanujan-type identities labelled by positive integers m and n. For fixed m and n the product side corresponds to a specialised character of the affine Kac-Moody algebra A_{2n}^{(2)} at level m, and is expressed as a product of n^2 theta functions of modulus 2m+2n+1, or by level-rank duality, as a product of m^2 theta functions. Rogers-Ramanujan-type identities for even moduli, corresponding to the affine Lie algebras C_n^{(1)} and D_{n+1}^{(2)}, are also proven.
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…
Recently, Rosengren utilized an integral method to prove a number of conjectural identities found by Kanade and Russell. Using this integral method, we give new proofs to some double sum identities of Rogers-Ramanujan type. These identities were earlier proved by approaches such as combinatorial arguments or by using $q$-difference equations. Our proofs are based on streamlined calculations, which relate these double sum identities to some known Rogers-Ramanujan type identities with single sums. Moreover, we prove a conjectural identity of Andrews and Uncu which was earlier confirmed by Chern.
Identities for the Rogers-Ramanujan Continued Fraction
本文通过 theta 函数剖分与五乘积恒等式证明了 Rogers–Ramanujan 连分数的若干新模恒等式，并由此得到 Rogers–Ramanujan 函数的新关系。
An Elliptic $BC_n$ Bailey Lemma, Multiple Rogers--Ramanujan Identities and Euler's Pentagonal Number Theorems
证明了经典的双参数Bailey引理的椭圆型$BC_n$推广，并得到了一个基本的单参数$BC_n$ Bailey引理的极限情形.作为应用，证明了与根系$BC_n$有关的几个求和和变换公式，包括$_6ϕ_5$求和公式、广义Watson变换和非专门化的Rogers-Selberg恒等式.最后一个恒等式被专门化以给出多边Roger-Selberg恒等式的无限族。标准的行列式评估，然后使用计算$B_n$和$D_n$推广的罗杰斯-拉马努金身份的决定因素的θ函数。从$BC_n$ $_6ϕ_5$求和公式出发，用一个类似的程序证明了一个无穷族的$D_n$欧拉五边形定理.
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。
The two Rogers-Ramanujan $q$-series \[ \sum_{n=0}^{\infty}\frac{q^{n(n+σ)}}{(1-q)\cdots (1-q^n)}, \] where $σ=0,1$, play many roles in mathematics and physics. By the Rogers-Ramanujan identities, they are essentially modular functions. Their quotient, the Rogers-Ramanujan continued fraction, has the special property that its singular values are algebraic integral units. We find a framework which extends the Rogers-Ramanujan identities to doubly-infinite families of $q$-series identities. If $a\in\{1,2\}$ and $m,n\geq 1$, then we have \[ \sum_{\substack{λλ_1\leq m}} q^{a|λ|} P_{2λ}(1,q,q^2,\dot…
At scattered places of his notebooks, Ramanujan recorded over 30 values of singular moduli $α_n$. All those results were proved by Berndt et. al by employing Weber-Ramanujan's class invariants. In this paper, we initiate to derive the explicit evaluations formula for $α_{9n}$ and $α_{n/9}$ by involving class invariant. For this purpose, we establish several new $P-Q$ mixed modular equations involving theta-functions. Further application of these modular equations, we derive a new formula to explicit evaluation of Ramanujan- Selberg continued fraction.
Non-Rascoe partitions and a rank parity function associated to the Rogers-Ramanujan partitions
研究Rogers-Ramanujan分拆中奇数秩与偶数秩超出数的生成函数，证明其与非Rascoe分拆（不同部分且部分数非部分）紧密相关，并推导算术性质及模4同余猜想，推广至参数ℓ。
Generalization of Ramanujan's Continued Fractions for Even Order
该研究借助拉马努金连分数恒等式推导偶数阶广义连分数，建立theta函数恒等式，以76阶连分数为例得到分拆理论恒等式等，推广了拉马努金连分数。
Let $B_{k,i}(n)$ be the number of partitions of $n$ with certain difference condition and let $A_{k,i}(n)$ be the number of partitions of $n$ with certain congruence condition. The Rogers-Ramanujan-Gordon theorem states that $B_{k,i}(n)=A_{k,i}(n)$. Lovejoy obtained an overpartition analogue of the Rogers-Ramanujan-Gordon theorem for the cases $i=1$ and $i=k$. We find an overpartition analogue of the Rogers-Ramanujan-Gordon theorem in the general case. Let $D_{k,i}(n)$ be the number of overpartitions of $n$ satisfying certain difference condition and $C_{k,i}(n)$ be the number of overpartition…
In this paper we refine a weighted partition identity of Alladi. We write explicit formulas of generating functions for the number of partitions grouped with respect to a partition statistic other than the norm. We tie our weighted results and the different statistics with the crank of a partition. In particular, we prove that the number of partitions into even number of distinct parts whose odd-indexed parts' sum is n is equal to the number of partitions of n with non-negative crank.
Eta-Quotient Representations for a Three Parameter Family of Modular Functions Associated with the Rogers-Ramanujan Continued Fraction
本文构造了与Rogers-Ramanujan连分数相关的三参数模函数族，推导其递归eta商表示，并应用于剖分公式，特别是用于获得受限奇数差分拆的5-剖分公式。
We present proofs of two new families of sum-product identities arising from the cylindric partitions paradigm. Most of the presented expressions, the related sum-product identities, and the ingredients for the proofs were first conjectured by Kanade-Russell in the spirit of Andrews-Schilling-Warnaar identities of the $A_2$ Rogers-Ramanujan type. We follow the footsteps of Kanade-Russell while we alter the computations heavily to accomplish our goals.
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。
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