A WZ proof of Ramanujan's Formula for Pi

We briefly review some of Ramanujan's contributions to mathematics, including his $1/\pi$ series, his work on modular forms, and his work on partitions. We briefly review his life, including his collaboration with Hardy. Finally, we give a brief summary of what any prospective mathematician should know about Ramanujan's work in number theory, including the rich relationship between his work on partitions and his work on modular forms. The title of this paper is a reference to...

In this article, we define functions analogous to Ramanujan's function $f(n)$ defined in his famous paper "Modular equations and approximations to $\pi$". We then use these new functions to study Ramanujan's series for $1/\pi$ associated with the classical, cubic and quartic bases.

In terms of the hypergeometric method, we give the extensions of two known series for $\pi$. Further, other twenty-nine summation formulas for $\pi$, $\pi^2$ and $1/\pi$ with free parameters are also derived in the same way.

Currently the circle and the sieve methods are the key tools in analytic number theory. In this paper the unifying theme of the two methods is shown to be the Ramanujan - Fourier series.

We use Zeilberger's algorithm for proving some identities of Ramanujan-type via $_2F_1$ evaluations.

We outline an elementary method for proving numerical hypergeometric identities, in particular, Ramanujan-type identities for $1/\pi$. The principal idea is using algebraic transformations of arithmetic hypergeometric series to translate non-singular points into singular ones, where the required constants can be computed using asymptotic analysis.

The author gives the full list of his conjectures on series for powers of $\pi$ and other important constants scattered in some of his public papers or his private diaries. The list contains 234 reasonable conjectural series. On the list there are 178 reasonable series for $\pi^{-1}$, four series for $\pi^2$, two series for $\pi^{-2}$, four series for $\pi^4$, two series for $\pi^5$, three series for $\pi^6$, seven series for $\zeta(3)$, one series for $\pi\zeta(3)$, two seri...

In this note, it is shown that the Ramanujan Master Theorem (RMT) when n is a positive integer can be obtained, as a special case, from a new integral formula. Furthermore, we give a simple proof of the RMT when n is not an integer.

In this paper, based on the WZ theory, a very succinct new proof, of an identity by Chaundy and Bullard, was given.

We find new hypergeometric identities which, in a certain aspect, are stron-ger than others of the same style found by the author in a previous paper. The identities in Section \ref{section-pi} are related to some Ramanujan-type series for $1/\pi$. We derive them by using WZ-pairs associated to some interesting formulas by Wenchang Chu. The identities we prove in Section \ref{section-pi2} are of the same style but related to Ramanujan-like series for $1/\pi^2$.