Related papers: Ramanujan series with a shift
In this article, we consider systems of linear congruences in several variables and obtain necessary and sufficient conditions as well as explicit expressions for the number of solutions subject to certain restriction conditions. These…
Using the theory of Calabi-Yau differential equations we obtain all the parameters of Ramanujan-Sato-like series for $1/\pi^2$ as $q$-functions valid in the complex plane. Then we use these q-functions together with a conjecture to find new…
We extend to an arbitrary number field the best known bounds towards the Ramanujan conjecture for the groups GL(n), n=2, 3, 4. In particular, we present a technique which overcomes the analytic obstacles posed by the presence of an infinite…
An asymptotic series in Ramanujan's second notebook (Entry 10, Chapter 3) is concerned with the behavior of the expected value of $\phi(X)$ for large $\lambda$ where $X$ is a Poisson random variable with mean $\lambda$ and $\phi$ is a…
We prove a kind of bilateral semi-terminating series related to Ramanujan-like series for negative powers of $\pi$, and conjecture a type of supercongruences associated to them. We support this conjecture by checking all the cases for many…
We generalize certain recent results of Ushiroya concerning Ramanujan expansions of arithmetic functions of two variables. We also show that some properties on expansions of arithmetic functions of one and several variables using classical…
A simple integration by parts and telescopic cancellation leads to a rigorous derivation of the first 2 terms for the error in Ramanujan's asymptotic series for the nth partial sum of the harmonic series. Then Kummer's transformation gives…
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.
Our main results are a WZ-proof of a new Ramanujan-like series for $1/\pi^2$ and a hypergeometric identity involving three series.
We study Ramanujan-Fourier series of certain arithmetic functions of two variables. We generalize Delange's theorem to the case of arithmetic functions of two variables and give sufficient conditions for pointwise convergence of…
A Ramanujan-type formula involving the squares of odd zeta values is obtained. The crucial part in obtaining such a result is to conceive the correct analogue of the Eisenstein series involved in Ramanujan's formula for $\zeta(2m+1)$. The…
A map is a panorama in small scale. In this half-survey, half-research paper we give general results on Ramanujan expansions. We don't include the ocean of results from the literature on the two classes (see Schwarz-Spilker Book, also…
In this note, we study the flint hills series of the form \begin{align} \sum \limits_{n=1}^{\infty}\frac{1}{(\sin^2n) n^3}\nonumber \end{align} via a certain method. The method essentially works by erecting certain pillars sufficiently…
We consider integral and series transformations, which are associated with Ramanujan's identities, involving various arithmetic functions and a ratio of products of Riemann's zeta functions of different arguments. Reciprocal inversion…
A conjectured relation between Ramanujan's asymptotic approximations to the exponential function and the exponential integral is established. The proof involves Stirling numbers, second-order Eulerian numbers, modifications of both of…
This paper aims to show that by making use of Ramanujan's Master Theorem and the properties of the lower incomplete gamma function, it is possible to construct a finite Mellin transform for the function $f(x)$ that has infinite series…
We prove an interesting symmetric $q$-series identity which generalizes a result due to Ramanujan. A proof that is analytic in nature is offered, and a bijective-type proof is also outlined.
In this paper, we introduce a new series of Rogers-Ramanujan-Gordon partitions when k = 3. The combinatorial interpretation of the series is given by base partition, forward moves and backward moves. We conclude the paper with future…
We prove that there is a correspondence between Ramanujan-type formulas for 1/\pi, and formulas for Dirichlet L-values. The same method also allows us to resolve certain values of the Epstein zeta function in terms of rapidly converging…
We generalize Watson's $ q $-analogue of Ramanujan's Entry 40 continued fraction by deriving solutions to a $ {}_{10} \phi_9 $ series contiguous relation and applying Pincherle's theorem. Watson's result is recovered as a special…