Jesus Guillera
We prove, by the WZ-method, some hypergeometric identities which relate ten extended Ramanujan type series to simpler hypergeometric series. The identities we are going to prove are valid for all the values of a parameter $a$ when they are convergent. Sometimes, even if they do not converge, they are valid if we consider these identities as limits.
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We give a simple unified proof for all existing rational hypergeometric Ramanujan identities for $1/π$, and give a complete survey (without proof) of several generalizations: rational hypergeometric identities for $1/π^c$, Taylor expansions, upside-down formulas, and supercongruences.
We find new hypergeometric identities which, in a certain aspect, are stron-ger than others of the same style found by the author in a previous paper. The identities in Section \ref{section-pi} are related to some Ramanujan-type series for $1/π$. We derive them by using WZ-pairs associated to some interesting formulas by Wenchang Chu. The identities we prove in Section \ref{section-pi2} are of the same style but related to Ramanujan-like series for $1/π^2$.
Eigenvectors of the discrete Fourier transform can be expressed using Ramanujan theta functions. New theta function identities, Ramanujan theta function identities, and generating functions for the quadratic numbers are a consequence.
Using Ramanujan's identities and the Weierstrass-Enneper representation of minimal surfaces and the analogue for Born-Infeld solitons, we derive further non-trivial identities.
Using the machinery from the theory of Calabi-Yau differential equations, we find formulas for $1/π^2$ of hypergeometric and non-hypergeometric types.
We give some generalizations to three identities of Srinivasa Ramanujan involving greatest integer function.
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。
Using a pair of two variable series-product identities recorded by Ramanujan in the lost notebook as inspiration, we find some new identities of similar type. Each identity immediately implies an infinite family of Rogers-Ramanujan type identities, some of which are well-known identities from the literature. We also use these identities to derive some general identities for integer partitions.
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 provide finite analogs of a pair of two-variable $q$-series identities from Ramanujan's lost notebook and a companion identity.
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。
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 derive two new generalizations of the Busche-Ramanujan identities involving the multiple Dirichlet convolution of arithmetic functions of several variables. The proofs use formal multiple Dirichlet series and properties of symmetric polynomials of several variables.
Using results from Ramanujan's lost notebook, Zudilin recently gave an insightful proof of a radial limit result of Folsom, Ono, and Rhoades for mock theta functions. Here we see that the author's previous work on the dual nature of Appell--Lerch sums and partial theta functions and on constructing bilateral $q$-series with mixed mock modular behaviour is well-suited for such radial limits. We present five more radial limit results which follow from mixed mock modular bilateral $q$-hypergeometric series. We also obtain the mixed mock modular bilateral series for a universal mock theta function…
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。
Infinite series are evaluated through the manipulation of a series for $\cos(2t \sin^{-1}x)$ resulting from Clausen's Product. Hypergeometric series equal to an expression involving $\frac{1} π$ are determined. Techniques to evaluate generalized hypergeometric series are discussed through perspectives of experimental mathematics.
For the unrestricted partition function $p(n)$ for integers $n \geq 0$, it is known that $p(49n + 19) \equiv 0 \pmod{49}$, $p(49n + 33) \equiv 0 \pmod{49}$, and $p(49n + 40) \equiv 0 \pmod{49}$ for all $n \geq 0$. We find witness identities for these Ramanujan congruences.
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 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…
We prove some Lambert series which were stated by Gosper without proof or reference. As an application, we shall evaluate integrals involving Ramanujan theta function $ψ(q)$. Furthermore, motivated by Ramanujan's identities for $qψ^4(q^2)$ and $\fr{ψ^3(q)}{ψ(q^3)}$, we shall evaluate the squares of $qψ^4(q^2)$ and $\fr{ψ^3(q)}{ψ(q^3)}$ in terms of Lambert series.
Rankin-Cohen Brackets and van der Pol-Type Identities for the Ramanujan's Tau Function
我们对模形式和拟模形式使用Rankin-Cohen括号给出了D. Lanphier和D. Niebur关于Ramanujan τ函数的van der Pol型恒等式。作为推论，我们得到了涉及除数函数的卷积和和同余关系。
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.
In 1914, Ramanujan gave a list of 17 identities expressing $1/π$ as linear combinations of values of hypergeometric functions at certain rational numbers. Since then, identities of similar nature have been discovered by many authors. Nowadays, one of the standard approaches to this kind of identities uses the theory of modular curves. In this paper, we will consider the case of Shimura curves and obtain Ramanujan-type formulas involving special values of hypergeometric functions and products of Gamma values. The product of Gamma values are related to periods of elliptic curves with complex mul…
An elementary proof of some Ramanujan-type identities
本文通过初等方法证明了将黎曼ζ函数在整数点的平方表示为涉及双曲函数、digamma函数和伯努利数等级数的恒等式，并修正了文本中的不准确之处和参考文献。
New convolution identities of hypergeometric Bernoulli polynomials are presented. Two different approaches to proving these identities are discussed, corresponding to the two equivalent definitions of hypergeometric Bernoulli polynomials as Appell sequences.
Short proofs of Ramanujan-like identities for the eighth order mock theta function $V_{0}(q)$
该研究利用Appell函数性质给出八阶仿θ函数$V_0(q)$的拉马努金型恒等式的简短证明，并参照相关经典结果提出恒等式的推广形式。
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 this article, we provide an application of hypergeometric evaluation identities, including a strange valuation of Gosper, to prove several supercongruences related to special valuations of truncated hypergeometric series. In particular, we prove a conjecture of van Hamme.
The joy of multisection: elementary approaches to Ramanujan's lacunary identities for Bernoulli numbers
本文用几何对称性与生成函数两种多重分段方法重新推导并推广 Ramanujan 的 Bernoulli 数缺项恒等式。
New proofs of two identities of Ramanujan
本文利用 $Γ_1(7)$ 的特定模函数与 Klein 的 $PSL(2,7)$ 射影表示，给出了拉马努金若干层为 $7$ 的 theta 函数恒等式的新证明，并由此推出 Berndt 和 Zhang 的四个恒等式。
Some Identities of Ramanujan's q-Continued Fractions of Order Fourteen and Twenty-Eight, and Vanishing Coefficients
本文从Ramanujan的一般连分数恒等式推导出十四阶和二十八阶连分数，建立theta函数恒等式，并推导色分区恒等式及消失系数结果。
A series of formula is presented that are all inspired by the Ramanujan Notebooks [6]. One of them appears in the notebooks II about Zeta(3). That formula inspired others that appeared in 1998, 2006 and 2009 on the author's website and later in literature [1][2][3]. New formulas for π and the Catalan constant are presented and a surprising series of approximations. A new set of identities is given for Eisenstein series. All of the formulas are conjectural since they were found experimentally. A new method for the computation of p(n) is presented.
In this paper, we prove supercongruence relations for truncated $N$-tuple sums of basic hypergeometric series. As an application, we give double, triple, and quadruple sum analogs of some Ramanujan-type supercongruences.
We prove hypergeometric type identities for a function defined in terms of quotients of the $p$-adic gamma function. We use these identities to prove a supercongruence conjecture of Rodriguez-Villegas between a truncated $_4F_3$ hypergeometric series and the Fourier coefficients of certain weight four modular form.
Ramanujan's and Lim's Identities and Harmonic Maass--Jacobi Forms
研究Lim对Ramanujan奇数zeta值恒等式的推广，构造Eisenstein级数经典Eichler积分的Jacobi类似物，证明其为奇异调和Maass--Jacobi形式，并建立非全纯部分、反演公式及算子性质。
We apply the method established in our previous work to derive a Chudnovsky-Ramanujan type formula for the Legendre family of elliptic curves. As a result, we prove two identities for $1/π$ in terms of hypergeometric functions.
On Ramanujan's Continued Fractions of Orders Five, Ten, and Twenty and Associated Lambert Series Identities
本文利用Jacobi theta函数、模拟theta函数及Ramanujan求和公式，建立了阶数二十的拉马努金连分数与阶数十、Rogers-Ramanujan连分数的新恒等式，并推导了Lambert级数与theta函数的关系，扩展了先前工作。
In this paper we highlight the connection between Ramanujan cubic polynomials (RCPs) and a class of polynomials, the Shanks cubic polynomials (SCPs), which generate cyclic cubic fields. In this way we provide a new characterization for RCPs and we express the zeros of any RCP in explicit form, using trigonometric functions. Moreover, we observe that a cyclic transform of period three permutes these zeros. As a consequence of these results we provide many new and beautiful identities. Finally we connect RCPs to Gaussian periods, finding a new identity, and we study some integer sequences relate…
We classify all the zeros and non-zero values of a family of hypergeometric series in the $p$-adic setting. These values of hypergeometric series in the $p$-adic setting lead to transformations of hypergeometric series in the $p$-adic setting which can be described as $p$-adic analogues of Kummer's and Pfaff's linear transformations on classical hypergeometric series. We also evaluate certain summation identities for hypergeometric series in the $p$-adic setting as well as Gaussian hypergeometric series.
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.
On Ramanujan's lost notebook and new tenth-order like identities for second-, sixth-, and eighth-order mock theta functions
该研究依托拉马努金遗失的笔记本，借助Hickerson和Mortenson的Appell函数性质，发现并证明了3个六阶仿theta函数的类十阶新恒等式，还补充了19个不同阶仿theta函数组合的类十阶恒等式。
One variable Generalization of five entries of Ramanujan and their finite analogue
本文提出了Bhoria等人恒等式的单变量推广及其有限类比，并据此推导出Ramanujan五个$q$-级数恒等式的单变量推广及相应有限类比。
Ramanujan wrote the following identity \begin{align*} \sqrt{2 \left(1 - \frac{1}{3^2}\right) \left(1 - \frac{1}{7^2}\right) \left(1 - \frac{1}{11^2}\right) \left(1 - \frac{1}{19^2}\right)} \ = \ \left(1 + \frac{1}{7}\right) \left(1 + \frac{1}{11}\right) \left(1 + \frac{1}{19}\right), \end{align*} on which Berndt asked "Is this an isolated result, or are there other identities of this type?". Rebák provided formulas that generate many similar identities and believed that the curious identity is related to the reciprocal of the Landau-Ramanujan constant. In a previous work, Chu examined necessar…
We prove hypergeometric type summation identities for a function defined in terms of quotients of the $p$-adic gamma function by counting points on certain families of hyperelliptic curves over $\mathbb{F}_{q}$. We also find certain special values of that function.
In this work we investigate the heat kernel of the Laplace--Beltrami operator on a rectangular torus and the according temperature distribution. We compute the minimum and the maximum of the temperature on rectangular tori of fixed area by means of Gauss' hypergeometric function $_2F_1$ and the elliptic modulus. In order to be able to do this, we employ a beautiful result of Ramanujan, connecting hypergeometric functions, the elliptic modulus and theta functions. Also, we investigate the temperature distribution of the heat kernel on hexagonal tori and use Ramanujan's corresponding theory of s…
A series of Ramanujan, two-term dilogarithm identities and some Lucas series
本文研究一类与Ramanujan遗失笔记中级数相关的初等级数，利用反双曲反正切、多对数及Clausen函数给出闭式，并证明新的双项二重对数恒等式，求出若干Lucas数级数的闭式。
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 …
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.
Two identities involving Cohen-Ramanujan expansions
本文推导了Jordan亏函数的科恩-拉马努金展开式，并给出了相关求和的渐近公式。
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 find involutions for three Rogers-Ramanujan-Gordon type identities obtained by Andrews on the generating functions for partitions with part difference and parity restrictions.
We prove a new Bailey-type transformation relating WP-Bailey pairs. We then use this transformation to derive a number of new 3- and 4-term transformation formulae between basic hypergeometric series.
An iterative approach toward hypergeometric accelerations
本研究受Chu的超几何加速思路启发，基于Wilf依赖Zeilberger算法导出非齐次差分方程的加速方法，提出迭代拓展方法，证明了多个通用常数的加速拉马努金型公式及相关新加速结果。
In this paper we prove some Ramanujan-type formulas for $1/π$ but without using the theory of modular forms. Instead we use the WZ-method created by H. Wilf and D. Zeilberger and find some hypergeometric functions in two variables which are second components of WZ-pairs that can be certified using Zeilberger's EKHAD package. These certificates have an additional property which allows us to get generalized Ramanujan-type series which are routinely proven by computer. We call these second hypergeometric components of the WZ-pairs generators. Finding generators seems a hard task but using a kind …
Quadratic Forms, Exact Covering Systems, and Product Identities for Theta Functions
本文建立整二次型与精确覆盖系统的联系，提出结构框架生成无穷多涉及拉马努金theta函数的乘积恒等式，统一解释二十二个Rogers-Ramanujan恒等式，并推导三元二次型相关新恒等式。
We outline an elementary method for proving numerical hypergeometric identities, in particular, Ramanujan-type identities for $1/π$. The principal idea is using algebraic transformations of arithmetic hypergeometric series to translate non-singular points into singular ones, where the required constants can be computed using asymptotic analysis.
An elementary treatment of Ramanujan's elementary continued fractions
本文研究拉马努金第二笔记本第12章中的初等连分数，这些连分数是其超几何与$q$-超几何连分数的前身，并给出初等处理。
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.
I present here a collection of formulas inspired from the Ramanujan Notebooks. These formulas were found using an experimental method based on three widely available symbolic computation programs: PARI-Gp, Maple and Mathematica. A new formula is presented for Zeta(5).
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。
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