Related papers: The number $\pi$ and summation by $SL(2,\mathbb Z)…

A family of original formulae for computing number PI and its proof are presented. An algorithm is proposed to validate the results of this new algorithm.

New formulas for 1/Pi^2 are found by transforming Guillera's formulas

In this paper we prove some new series for $1/\pi$ as well as related congruences. We also raise several new kinds of series for $1/\pi$ and present some related conjectural congruences involving representations of primes by binary…

We show how to find series expansions for $\pi$ of the form $\pi=\sum_{n=0}^\infty {S(n)}\big/{\binom{mn}{pn}a^n}$, where S(n) is some polynomial in $n$ (depending on $m,p,a$). We prove that there exist such expansions for $m=8k$, $p=4k$,…

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…

By employing certain extended classical summation theorems, several surprising \pi and other formulae are displayed.

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.

Based on the framework of the WZ theory, a new evaluation for $\varsigma (2) = \frac{\pi ^2}{6}$ and $\varsigma (4) = \frac{\pi ^4}{90}$ was given respectively, finally, a new recurrence formula for $\varsigma (2k)$ was given.

By means of the telescoping method, we establish two sum- mation formulas on sine function. As the special cases of them, several interesting series expansions for $1/\pi^m$ and $\pi^m$.

Using techniques from calculus, we combine classical identities for $\pi$, $\operatorname{ln}2$, and harmonic numbers, to arrive at a nice infinite series formula for $\pi/3$ that does not appear to be well known. In addition, we give…

We present a new variant of the Faa di Bruno formula with a simpler summation order.

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

In our previous publication we have shown a method for calculating series of even powers of $\pi$ based on the product representation of the $sinc$ function. We refer the readers to [1] for more details. In this work we apply the method to…

An algorithm for computing /pi(N) is presented.It is shown that using a symmetry of natural numbers we can easily compute /pi(N).This method relies on the fact that counting the number of odd composites not exceeding N suffices to calculate…

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 $\pi$.

For every natural number $T,$ we write $\Ln T$ as a series, generalizing the known series for $\Ln 2.$

There are several reformulations of the Vi\`ete's formula for pi that have been reported in the modern literature. In this paper we show another analog to the Vi\`ete's formula for pi by Chebyshev polynomials of the first kind.

With the help of the partial derivative operator and several summation formulas for hypergeometric series, we find three double series for $\pi$. In terms of the operator just stated and several summation formulas for basic hypergeometric…

By some SL(2, Z) modular forms introduced in [11] and [4] , we get some interesting anomaly cancellation formulas. As corollaries, we get some divisibility results of index of twisted Dirac operators.

We derive a BBP-type formula for the remainder of the Madhava-Gregory-Leibniz series for $\pi$. The result is a closed form in base-$16$ with Pochhammer denominators. The analogous formula for the alternating series for $\log 2$ is also…