Related papers: Identities inspired by the Ramanujan Notebooks, se…
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…
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…
Recently, Simon Plouffe has discovered a number of identities for the Riemann zeta function at odd integer values. These identities are obtained numerically and are inspired by a prototypical series for Apery's constant given by Ramanujan:…
We revisit several entries from Ramanujan's notebooks which follow from more elementary arguments than a first glance may suggest. Our goal is to demystify these results through more accessible proofs, while also shining some light on the…
Fundamental mathematical constants like $e$ and $\pi$ are ubiquitous in diverse fields of science, from abstract mathematics to physics, biology and chemistry. For centuries, new formulas relating fundamental constants have been scarce and…
The Euler product for the Landau--Ramanujan constant could have motivated a curious identity by Ramanujan that appears in his notebooks two times. This observation involves a square root and the first four prime numbers of the form $4n+3$,…
The Ramanujan Machine project detects new expressions related to constants of interest, such as $\zeta$ function values, $\gamma$ and algebraic numbers (to name a few). In particular the project lists a number of conjectures concerning the…
In this note, by making use of a known hypergeometric series identity, I prove two Ramanujan-type series for the Catalan's constant. The convergence rate of these central binomial series surpasses those of all known similar series,…
In 2015 Cristian-Silviu Radu designed an algorithm to detect identities of a class studied by Ramanujan and Kolberg. This class includes the famous identities by Ramanujan which provide a witness to the divisibility properties of $p(5n+4),$…
In Ramanujan's Lost Notebook there is an amazing identity that furnishes infinitely many "almost counterexamples" to the cubic Fermat's Last Theorem, with no indication whatsoever how he discovered it. In 1995, Michael Hirschhorn explained,…
Ramanujan listed several q-series identities in his lost notebook. The most well known q-series identities are the Rogers-Ramanujan type identities which are first discovered by Rogers and then rediscovered by Ramanujan. In this paper, we…
Ramanujan stated an identity to the effect that if three sequences $\{a_n\}$, $\{b_n\}$ and $\{c_n\}$ are defined by $r_1(x)=:\sum_{n=0}^{\infty}a_nx^n$, $r_2(x)=:\sum_{n=0}^{\infty}b_nx^n$ and $r_3(x)=:\sum_{n=0}^{\infty}c_nx^n$ (here each…
Eisenstein series play an important role in the theory of modular forms and have profound connections with $q$-series identities, partition theory, and special functions. Likewise, Ramanujan's mock theta functions, originally introduced in…
In his lost notebook, Ramanujan listed 5 identities related to the false theta function $$f(q)=\sum_{n=0}^\infty (-1)^nq^{n(n+1)/2}.$$ A new combinatorial interpretation and proof of one of these identities is given. The methods of the…
We report new hypergeometric constructions of rational approximations to Catalan's constant, $\log2$, and $\pi^2$, their connection with already known ones, and underlying "permutation group" structures. Our principal arithmetic achievement…
In his second notebook, Ramanujan discovered the following identity for the special values of $\zeta(s)$ at the odd positive integers \begin{equation*}\begin{aligned}\alpha^{-m}\,\left\{\dfrac{1}{2}\,\zeta(2m + 1) + \sum_{n =…
One of the celebrated formulas of Ramanujan is about odd zeta values, which has been studied by many mathematicians over the years. A notable extension was given by Grosswald in 1972. Following Ramanujan's idea, we rediscovered a…
By employing the classical tools from the theory of $q$-series and theta functions, new fascinating identities on different continued fractions can be achieved. In this article, we use the product expansion of Jacobi's theta function to…
Our main results are a WZ-proof of a new Ramanujan-like series for $1/\pi^2$ and a hypergeometric identity involving three series.
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…