Arjun K. Rathie
In this short research note, we aim to establish an interesting extension of a summation due to Ramanujan.The result is derived with the help of an extension of Gauss's summation theorem available in the literature.
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In this paper, we give some extensions for Ramanujan's circular summation formula with the mixed products of two Jacobi's theta functions. As some applications, we also obtain many interesting identities of Jacobi's theta functions.
Using Abel summation the paper proves a weak form of the Wiener-Khinchin formula for arithmetic functions with point-wise convergent Ramanujan-Fourier expansions. The main result is that the convolution of most arithmetic functions possessing an R-F expansion are Abel-summable to a result involving only the Ramanujan-Fourier coefficients of the R-F expansion(s).
We use a specialization of Ramanujan's ${}_1ψ_1$ summation to give a new proof of a recent formula of Hickerson and Mortenson which expands a special family of Hecke-type double sums in terms of Appell-Lerch sums and theta functions.
In this short note, we aim to discuss some summations due to Ramanujan, their generalizations and some allied series
In this paper we obtain some new transformation formula for Ramanujan summation formula and also establish some eta-function identities. we also deduce a q-Gamma function identity, n q-integral and some interesting series representation.
In the sixth chapter of his notebooks Ramanujan introduced a method of summing divergent series which assigns to the series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula to the partial sums of the series. This method is now called the Ramanujan summation process. In this paper we calculate the Ramanujan sum of the exponential generating functions $\sum_{n\geq 1}\log n e^{nz}$ and $\sum_{n\geq 1}H_n^{(j)} e^{-nz}$ where $H_n^{(j)}=\sum_{m=1}^n \frac{1}{m^j}$. We find a surprising relation between the two sums when $j=1$ from w…
This paper gives a short but reasonably comprehensive review of Ramanujan's {_1ψ_1} summation and its generalisations. It covers the history of Ramanujan's summation, simple applications to sums of squares and orthogonal polynomials, non-commutative generalisations, and generalisations to affine root systems.
The neglected Russian mathematician, N.~S.~Koshliakov, derived beautiful generalizations of the classical Abel--Plana summation formula through a setting arising from a boundary value problem in heat conduction. When we let the parameter $p$ in this setting tend to infinity, his formulas reduce to the classical Abel--Plana summation formula. Rigorous formulations and proofs of these summation formulas are given. In his notebooks, Ramanujan derived different analogues of the Abel--Plana summation formula. One particular example provides a vast new generalization of the classical transformation …
Lucy Slater used Bailey's $_6ψ_6$ summation formula to derive the Bailey pairs she used to construct her famous list of 130 identities of the Rogers-Ramanujan type. In the present paper we apply the same techniques to Chu's $_{10}ψ_{10}$ generalization of Bailey's formula to produce quite general Bailey pairs. Slater's Bailey pairs are then recovered as special limiting cases of these more general pairs. In re-examining Slater's work, we find that her Bailey pairs are, for the most part, special cases of more general Bailey pairs containing one or more free parameters. Further, we also find ne…
Recently, $4$-regular partitions into distinct parts are connected with a family of overpartitions. In this paper, we provide a uniform extension of two relations due to Andrews for the two types of partitions. Such an extension is made possible with recourse to a new trivariate Rogers--Ramanujan type identity, which concerns a family of quadruple summations appearing as generating functions for the aforementioned overpartitions. More interestingly, the derivation of this Rogers--Ramanujan type identity is relevant to a certain well-poised basic hypergeometric series.
A smooth summation of Ramanujan expansions
本文提出Ramanujan级数的光滑求和概念，证明其在Wintner假设下收敛，并应用于相关性与Hardy–Littlewood孪生素数猜想。
An elliptic $BC_n$ generalization of the classical two parameter Bailey Lemma is proved, and a basic one parameter $BC_n$ Bailey Lemma is obtained as a limiting case. Several summation and transformation formulas associated with the root system $BC_n$ are proved as applications, including a $_6ϕ_5$ summation formula, a generalized Watson transformation and an unspecialized Rogers--Selberg identity. The last identity is specialized to give an infinite family of multilateral Rogers--Selberg identities. Standard determinant evaluations are then used to compute $B_n$ and $D_n$ generalizations of t…
In terms of the hypergeometric method, we establish the extensions of two formulas for $1/π$ due to Ramanujan [27]. Further, other five summation formulas for $1/π$ with free parameters are also derived in the same way.
It is mathematical folklore that 1 + 2 + 3 + 4 + ... = --1/12. This result is usually achieved using elaborate analytical methods, such as zeta function regularization or Ramanujan summation. However, in its notebooks, Ramanujan has also provided a very simple derivation which relied instead on algebraic manipulations. Recently, a video from Numberphile has presented a similar derivation of the result (provoking lots of discussions and debates about the meaning of such an equality). But this derivation, simple as it is, is usually considered as less rigorous than those using more elaborate ana…
In this paper, we introduce some explicit approximations for the summation $\sum_{k\leq n}Ω(k)$, where $Ω(k)$ is the total number of prime factors of $k$.
We continue the analysis of the elliptic part of the trace formula initiated in \cite{Altug:2015aa}. In that reference Poisson summation was applied to the elliptic part and the dominant term was analyzed. The main aim of this paper is to study the remaining terms after Poisson summation. We analyze the the Fourier transforms of (smoothed) orbital integrals and obtain exact asymptotic expansions. As an application we recover, using the Arthur-Selberg trace formula, Kuznetsov's result (cf. \cite{Kuznetsov:1980aa}) that the trace of the $p$th Hecke operator on cuspidal automorphic representation…
Recently Lachterman, Schayer, and Younger published an elegant proof of the Ramanujan congruences for the partition function $p(n)$. Their proof uses only the classical theory of modular forms as well as a beautiful result of Choie, Kohnen, and Ono, without need for Hecke operators. In this paper we give a method for generalizing Lachterman, Schayer, and Younger's proof to include Ramanujan congruences for multipartition functions $p_k(n)$, and Ramanujan congruences for $p(n)$ modulo certain prime powers.
Voronoi summation formulas, oscillations of Riesz sums, and Ramanujan-Guinand and Cohen type identities
本研究推导了刘维尔函数、莫比乌斯函数及除数函数平方的Voronoi求和公式，得到多类恒等式特例，并在相关数论猜想假设下证明了对应Riesz和及其误差项的振荡性质。
We define a new kind of classical digamma function, and establish its some fundamental identities. Then we apply the formulas obtained, and extend tools developed by Flajolet and Salvy to study more general Euler type sums. The main results of Flajolet and Salvy's paper \cite{FS1998} are the immediate corollaries of main results in this paper. Furthermore, we provide some parameterized extensions of Ramanujan-type identities that involve hyperbolic series. Some interesting new consequences and illustrative examples are considered.
Let $k\geq 2$ be an integer and $j$ an integer satisfying $1\leq j \leq 4k-5$. We define a family $\{ C_{j,k}(z) \}_{1\leq j \leq 4k-5} $ of eta quotients, and prove that this family constitute a basis for the space $S_{2k} (Γ_0 (12))$ of cusp forms of weight $2k$ and level $12$. We then use this basis together with certain properties of modular forms at their cusps to prove an extension of the Ramanujan-Mordell formula.
On certain extensions of Ramanujan's Master Theorem and their applications
本文基于 k-gamma 函数与 p-k gamma 函数推广 Ramanujan 主定理，建立 Mellin 型二重积分及推论，并用于计算多种定积分。
We use properties of modular forms to prove the following extension of the Ramanujan-Mordell formula, \begin{align*} z^{k-j}z_p^{j}=&\frac{p_χ^{k-j}-1}{p_χ^{k}-1}F_p(k,j;τ)+ \frac{p_χ^{k}-p_χ^{k-j}}{p_χ^{k}-1}F_p(k,j;pτ)+z^{k} A_p(k,j;τ), \end{align*} for all $ 1 < k \in \mathbb{n} $, $0 \leq j \leq k$ and $p$ an odd prime. We obtain this result by computing the Fourier series expansions of modular forms at all cusps of $Γ_0(4p)$.
If $f(z)$ is a modular form of weight $k$, then the differential operator $\vartheta_k$ defined by $\vartheta_k(f) = \frac{1}{2πi} \frac{d}{dz}f(z) - \frac{k}{12} E_2(z) f(z)$ (known as the Ramanujan-Serre derivative map) is a modular form of weight $k+2$. In this paper, we obtain a simple extension of this map and use it to get a general method to derive certain convolution sums of the divisor functions (using the theory of modular forms). Explicit expressions are given for four types of convolution sums and we provide many examples for all these types of sums.
If a continued fraction $K_{n=1}^{\infty} a_{n}/b_{n}$ is known to converge but its limit is not easy to determine, it may be easier to use an extension of $K_{n=1}^{\infty}a_{n}/b_{n}$ to find the limit. By an extension of $K_{n=1}^{\infty} a_{n}/b_{n}$ we mean a continued fraction $K_{n=1}^{\infty} c_{n}/d_{n}$ whose odd or even part is $K_{n=1}^{\infty} a_{n}/b_{n}$. One can then possibly find the limit in one of three ways: (i) Prove the extension converges and find its limit; (ii) Prove the extension converges and find the limit of the other contraction (for example, the odd part, if $K_{…
A number field extension of Ramanujan's identity and its integral representation via the generalized Riemann $Ξ$-function
本文建立拉马努金恒等式的数域类似物，得到戴德金ζ函数高次幂变换公式的积分表示，重现Hardy的雅可比θ关系积分表示，还得到扩展拉马努金-科什利亚科夫公式的数域类似物。
This paper presents a family of rapidly convergent summation formulas for various finite sums of the form $\sum_{k=0}^{\lfloor x\rfloor}f(k)$, where $x$ is a positive real number.
General summation formulas have been proved to be very useful in analysis, number theory and other branches of mathematics. The Lipschitz summation formula is one of them. In this paper, we give its application by providing a new transformation formula which generalizes that of Ramanujan. Ramanujan's result, in turn, is a generalization of the modular transformation of Eisenstein series $E_k(z)$ on SL$_2(\mathbb{Z})$, where $z\to-1/z, z\in\mathbb{H}$. The proof of our result involves delicate analysis containing Cauchy Principal Value integrals. A simpler proof of a recent result of ours with …
On a new modular equation of degree five and Eisenstein series identities of associated levels
本文利用拉曼纽尔的$_1ψ_1$求和公式获得level十的theta函数恒等式，推导出五次模方程，并通过巴伊的$_6ψ_6$求和公式建立level十的埃西尔斯坦级数恒等式。
We define an extension of the Ramanujan trigonometric function to arbitrary dimensions, and give the Dirichlet series generating function. The extension was first given by Eckford Cohen long ago. This links directly to visible point vector identities, and possibly to lattice sums in Physics and Chemistry presented by Baake et al. New generating functions and summations are given here, generalizing the Ramanujan function, Euler totient and the Jordan totient functions, based on visible lattice point ideas.
An extension of Ramanujan-Guinand identity for the Dedekind zeta function and a new formula for $ζ_\mathbb{F}(2)$ and $ζ_\mathbb{F}(2m+1)$
本文为戴德金ζ函数建立拉马努金-吉南恒等式的数域类似物，得到ζ_F(2)的新公式及ζ_F(2m+1)的恒等式，还获戴德金η函数对数的变换公式。
The paper consists of two parts. The aim of the first, and main, part is to explain, in an elementary way, Hasse's proof of Ramanujan-Nagell's Theorem. In the second part, we formulate some natural extensions of Ramanujan-Nagell's equation.
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。
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。
We consider an extension of the Ramanujan series with a variable $x$. If we let $x=x_0$, we call the resulting series: "Ramanujan series with the shift $x_0$". Then, we relate these shifted series to some $q$-series and solve the case of level $4$ with the shift $x_0=1/2$. Finally, we indicate a possible way towards proving some patterns observed by the author corresponding to the levels $\ell=1, 2, 3$ and the shift $x_0=1/2$.
Ramanujan provided several results involving the modified Bessel function $K_z(x)$ in his Lost Notebook. One of them is the famous Ramanujan-Guinand formula, equivalent to the functional equation of the non-holomorphic Eiesenstien series on $SL_2(z)$. Recently, this formula was generalized by Dixit, Kesarwani, and Moll. In this article, we first obtain a generalization of a theorem of Watson and, as an application of it, give a new proof of the result of Dixit, Kesarwani, and Moll. Watson's theorem is also generalized in a different direction using ${}_μK_z(x,λ)$ which is itself a generalizati…
Ramanujan's reciprocity theorem may be considered as a three-variable extension of Jacobi's triple product identity. Using the method of $q$-partial differential equations, we extend Ramanujan's reciprocity theorem to a seven-variable reciprocity formula. The Andrews--Askey integral is a $q$-integral having four parameters with base $q$. Using the same method we extend the Andrews--Askey integral formula to a $q$-integral formula which has seven parameters with base $q$.
An extension of the Chudnovsky algorithm
本研究借助Chudnovsky级数的无穷推广与Gauss-Salamin-Brent迭代，提出一种每一项可产生约153位数字的高阶Chudnovsky算法扩展，优于当前每一项110位的世界纪录，并给出PARI/GP实现方案。
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。
In a recent work, Gun and co-workers have proposed that $\,\sum_{n=-\infty}^{\infty}{(n+α)^{-k}}\,$ is a transcendental number for all integer $\,k$, $k > 1$, and $\,α\in \mathbb{Q} \backslash \mathbb{Z}$. Here in this work, this proposition is shown to be \emph{false} whenever $\,k\,$ is odd and $\,α\,$ is a half-integer. It is also shown that these are the only counterexamples, which allows for a correct reformulation of the original proposition. This leads to a theorem yielding a closed-form expression for the summation of that series, which determines its arithmetic nature. The result is t…
Two inequalities concerning the symmetry of the zeta-function and the Ramanujan $τ$-function are improved through the use of some elementary considerations.
Further Extensions of Sury's Identity
本文通过简单方法扩展Sury恒等式，推导出多个新的求和公式，连接卢卡斯数、斐波那契数和2的幂。
Parameterized partial theta identities and a unified $q$-difference proof
本文通过 Andrews 变换的双参数特化，以 $q$-差分递推和边界估计统一证明了 Ramanujan 部分 theta 恒等式的五族整数参数扩展，并应用于 Lovejoy 残差恒等式，构造共轭 Bailey 对。
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