Related papers: One-parameter Generalizations of Ramanujan's Formu…

In this article we use theoretical and numerical methods to evaluate in a closed-exact form the parameters of Ramanujan type $1/\pi$ formulas.

Ramanujan's series for Pi, that appeared in his famous letter to Hardy, is given a one-line WZ proof.

In this article we give the theoretical background for generating Ramanujan type $1/\pi^{2\nu}$ formulas. As applications of our method we give a general construction of $1/\pi^4$ series and examples of $1/\pi^6$ series. We also study the…

In terms of the hypergeometric method, we establish the extensions of two formulas for $1/\pi$ due to Ramanujan [27]. Further, other five summation formulas for $1/\pi$ with free parameters are also derived in the same way.

In this paper we want to prove some formulas listed by S. Ramanujan in his paper "Modular equations and approximations to $\pi$" \cite{24} with an elementary method.

The known WZ-proofs for Ramanujan-type series related to $1/\pi$ gave us the insight to develop a new proof strategy based on the WZ-method. Using this approach we are able to find more generalizations and discover first WZ-proofs for…

We use a variant of Wan's method to prove two Ramanujan-Orr type formulas for $1/\pi$. This variant needs to know in advance the formulas for $1/\pi$ that we want to prove, but avoids the need of solving a system of equations.

In a famous paper of $1914$ Ramanujan gave a list of $17$ extraordinary formulas for the number $\pi$. In this paper we explain a general method to prove them, based on an original idea of James Wan and in some own ideas.

Explicit formulas involving a generalized Ramanujan sum are derived. An analogue of the prime number theorem is obtained and equivalences of the Riemann hypothesis are shown. Finally, explicit formulas of Bartz are generalized.

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 work, we establish modular parameterizations for two general formulas for $\frac{1}{\pi}$ that subsume conjectural Ramanujan type formulas due to Z.-W. Sun, which have remained open since 2011. As an application of this, in a…

Using the WZ-method we find some of the easiest Ramanujan's formulae and also some new interesting Ramanujan-like sums.

In 1914 S. Ramanujan recorded a list of 17 series for $1/\pi$. We survey the methods of proofs of Ramanujan's formulae and indicate recently discovered generalizations, some of which are not yet proven.

Using some properties of the gamma function and the well-known Gauss summation formula for the classical hypergeometric series, we prove a four-parameter series expansion formula, which can produce infinitely many Ramanujan type series for…

We prove q-analogues of two Ramanujan-type series for $1/\pi$ from $q$-analogues of ordinary WZ pairs.

We have found several summation formulas that extend Ramanujan's psi sum. First contains a parameter $\alpha=1/N$, $N$ is a positive integer, and transforms to $q$-beta integral in the limit $N\to\infty$. The other is a $q$-analogue of…

We explain the use and set grounds about applicability of algebraic transformations of arithmetic hypergeometric series for proving Ramanujan's formulae for $1/\pi$ and their generalisations.

We prove two new series of Ramanujan type for $1/\pi^2$.

We prove some "divergent" Ramanujan-type series for $1/\pi$ and $1/\pi^2$ applying a Barnes-integrals strategy of the WZ-method.

We give a simple unified proof for all existing rational hypergeometric Ramanujan identities for $1/\pi$, and give a complete survey (without proof) of several generalizations: rational hypergeometric identities for $1/\pi^c$, Taylor…