相关论文: A WZ proof of Ramanujan's Formula for Pi

Several terminating generalizations of Ramanujan's formula for $\frac{1}{\pi}$ with complete WZ proofs are given.

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…

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.

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

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

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

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.

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.

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 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…

First we give general formulas for proving real or complex Ramanujan series for $1/\pi$. Then, as an example, we apply them for providing complete proofs of the fastest series for $1/\pi$ due to Ramanujan using Russell and Weber modular…

We show with some examples how to prove some Ramanujan-type series for $1/\pi$ in an elementary way by using terminating identities.

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…

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

This is an elementary explanation of a cubic composition formula due to Ramanujan.

In 1914, Ramanujan presented a collection of 17 elegant and rapidly converging formulae for $\pi$. Among these, one of the most celebrated is the following series:…

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.

We make a summary of the different types of proofs adding some new ideas. In addition we conjecture some relations which could be necessary in "modular type proofs" (not still found) of the Ramanujan-like series for 1/\pi^2.

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