K. Srinivasa Rao, G. Vanden Berghe, Christian Krattenthaler
Example 7, after Entry 43, in Chapter XII of the first Notebook of Srinivasa Ramanujan is proved and, more generally, a summation theorem for $_3F_2(a,a,x;1+a,1+a+N;1)$, where $N$ is a non-negative integer, is derived.
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In his third notebook, Ramanujan claims that $$ \int_0^\infty \frac{\cos(nx)}{x^2+1} \log x \,\mathrm{d} x + \fracπ{2} \int_0^\infty \frac{\sin(nx)}{x^2+1} \mathrm{d} x = 0. $$ In a following cryptic line, which only became visible in a recent reproduction of Ramanujan's notebooks, Ramanujan indicates that a similar relation exists if $\log x$ were replaced by $\log^2x$ in the first integral and $\log x$ were inserted in the integrand of the second integral. One of the goals of the present paper is to prove this claim by contour integration. We further establish general theorems similarly rela…
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).
We study a continued fraction due to Ramanujan, that he recorded as Entry 12 in Chapter 16 of his second notebook. It is presented in Part III of Berndt's volumes on Ramanujan's notebooks. We give two alternate approaches to proving Ramanujan's Entry 12, one using a method of Euler, and another using the theory of orthogonal polynomials. We consider a natural generalization of Entry 12 suggested by the theory of orthogonal polynomials.
Generalizations (in the spirit of Koshliakov) of some formulas from Ramanujan's Lost Notebook
本文在科什利亚科夫理论框架下推广了拉马努金遗失笔记中关于除数函数和戴德金η函数对数变换的若干恒等式。
We generalize Watson's $ q $-analogue of Ramanujan's Entry 40 continued fraction by deriving solutions to a $ {}_{10} ϕ_9 $ series contiguous relation and applying Pincherle's theorem. Watson's result is recovered as a special terminating case, while a limit case yields a new continued fraction associated with an $ \ephis $ series contiguous relation.
Ramanujan's $q$-continued fractions are a central part of Ramanujan's development of basic hypergeometric series. They appear in Chapter 16 of Part III and Chapter 32 of Part V of {\em Ramanujan's Notebooks} edited by Berndt, and in Volume I of Andrews and Berndt's {\em Ramanujan's Lost Notebook}. In these references the continued fractions as presented in the order in which they appear in Ramanujan's original notebooks. We summarize the work of several authors on this topic and re-organize Ramanujan's $q$-continued fractions.
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。
We provide a rigorous formulation of Entry 17(v) in Ramanujan's Notebooks and show how this relates to the first Stieltjes constant. A new proposition 4.5 is included to show the close relationship with some analysis presented by Candelpergher in 2017.
In Entry 16, Chapter 16 of his notebooks, Ramanujan himself gave a formula for the convergents of the famous Rogers-Ramanujan continued fraction. We provide a similar formula for the convergents of a more general continued fraction, namely Entry 15 of Chapter 16.
A Ramanujan enigma: Part 2
本文严格处理Hardy著作中提及的正弦函数发散级数，并重新审视Ramanujan笔记本中的条目17(v)。
Period Function of Maass forms from Ramanujan's Lost Notebook
本文证明拉马努金失散笔记第220页的函数$\mathcal{F}_1(x)$满足Maass形式的周期关系，称为拉马努金周期函数，并发现其为Hecke本征函数，推广至参数$s$，连接更广泛的周期函数理论。
Ninth degree analogue of Ramanujan's septic theta function identity
本文提出了Ramanujan七次θ函数恒等式的九次类似，并应用其笔记簿中的三次方程条目、用θ函数解释三角恒等式，最后计算了五个φ(e^{-π√n})的例子。
Throughout his entire mathematical life, Ramanujan loved to evaluate definite integrals. One can find them in his problems submitted to the \emph{Journal of the Indian Mathematical Society}, notebooks, Quarterly Reports to the University of Madras, letters to Hardy, published papers and the Lost Notebook. His evaluations are often surprising, beautiful, elegant, and useful in other mathematical contexts. He also discovered general methods for evaluating and approximating integrals. A survey of Ramanujan's contributions to the evaluation of integrals is given, with examples provided from each o…
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。
Cubic and quintic analogues of Ramanujan's septic theta function identity
受Ramanujan七次θ函数恒等式补全的启发，本文提出三次和五次类似，并结合Ramanujan类不变量，为其著名θ函数φ(q)提供了多种求值。
On a function of Ramanujan twisted by a logarithm
本文利用拉马努金《遗失笔记》中的模关系和德宁格函数导数的近期结果，推导出涉及双伽马函数与对数因子的无穷级数函数方程，并研究了其中起关键作用的积分 $\mathscr{H}(x)$ 的多种表示。
An asymptotic series in Ramanujan's second notebook (Entry 10, Chapter 3) is concerned with the behavior of the expected value of $ϕ(X)$ for large $λ$ where $X$ is a Poisson random variable with mean $λ$ and $ϕ$ is a function satisfying certain growth conditions. We generalize this by studying the asymptotics of the expected value of $ϕ(X)$ when the distribution of $X$ belongs to a suitable family indexed by a convolution parameter. Examples include the problem of inverse moments for distribution families such as the binomial or the negative binomial.
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…
The three missing terms in Ramanujan's septic theta function identity
本文给出了 Ramanujan 遗失笔记簿中关于其 theta 函数 $φ(q)$ 的七次恒等式里缺失的三个具体项，并进一步计算了类不变量 $G_{343}$ 及 $φ(e^{-nπ})$ 在特定整数下的特殊值。
An asymptotic expansion inspired by Ramanujan
本文验证了 Ramanujan 笔记本中关于一个和式渐近于 ln(x)+γ 的论断在 n=2 时正确，并讨论了误差项阶数及 n=1 情况的不同正确推广。
Motivated by Lévy's characterization of Brownian motion on the line, we propose an analogue of Brownian motion that has as its state space an arbitrary closed subset of the line that is unbounded above and below: such a process will be a martingale, will have the identity function as its quadratic variation process, and will be ``continuous'' in the sense that its sample paths don't skip over points. We show that there is a unique such process, which turns out to be automatically a reversible Feller-Dynkin Markov process. We find its generator, which is a natural generalization of the operator…
It is pointed out that the generalized Lambert series $\displaystyle\sum_{n=1}^{\infty}\frac{n^{N-2h}}{e^{n^{N}x}-1}$ studied by Kanemitsu, Tanigawa and Yoshimoto can be found on page $332$ of Ramanujan's Lost Notebook in a slightly more general form. We extend an important transformation of this series obtained by Kanemitsu, Tanigawa and Yoshimoto by removing restrictions on the parameters $N$ and $h$ that they impose. From our extension we deduce a beautiful new generalization of Ramanujan's famous formula for odd zeta values which, for $N$ odd and $m>0$, gives a relation between $ζ(2m+1)$ a…
Modular relations involving generalized digamma functions
本文针对广义 digamma 函数建立了模关系，推广了 Ramanujan 的著名变换，并推导了 Carlitz 型和 Guinand 型有限模关系，其中涉及组合和与反演公式。核心贡献在于对任意 k 给出模关系及渐近展开，并处理了微妙情形。
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