A WZ proof of Ramanujan's Formula for Pi

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.

The document contains an outline of a modular proof for Ramanujan-Chudnovsky identity.

Re presenting the traditional proof of Srinivasa Ramanujan's own favorite series for the reciprocal of $\pi$ :\begin{equation}\frac{1}{\pi} = \frac{\sqrt{8}}{9801} \sum_{n=0}^{+\infty} \frac{(4n)!}{(n!)^4} \frac{1103 + 26390n}{396^{4n}} \; \text{,}\end{equation}as well as several other examples of Ramanujan's infinite series. As a matter of fact, the derivation of such formulae has involved specialized knowledge of identities of classical functions and modular functions.

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

Recently Z.W.Sun found over hundred conjectured formulas for 1/pi. Many of them were proved by H.H.Chan, J.Wan andW.Zudilin (see [3], [8] in the paper). Here we show that several other formulas in [6] are simple transformations of known formulas for 1/pi.

In this Note, we start off with the primary representation of e and from there present an elementary short proof for the Wallis formula for $\pi$.

It will be shown a different way to find infinite series for Pi involving complex conjugates.

By applying the derivative operator to the known identities from hypergeometric series or WZ pairs, we obtain seven series associated with harmonic numbers. Specifically, six of them are Ramanujan-like formulas for $1/\pi$ and the remaining onecontains harmonic numbers of order $2$. As conclusions, Sun's five conjectural series are proved.

In this paper we prove some Ramanujan-type formulas for $1/\pi$ but without using the theory of modular forms. Instead we use the WZ-method created by H. Wilf and D. Zeilberger and find some hypergeometric functions in two variables which are second components of WZ-pairs that can be certified using Zeilberger's EKHAD package. These certificates have an additional property which allows us to get generalized Ramanujan-type series which are routinely proven by computer. We call...

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