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We derive, using a heuristic method, a $p$-adic mate of bilateral Ramanujan series. It has (among other consequences) the Zudiln's supercongruences for rational Ramanujan series., Comment: 9 pages

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 expansions, upside-down formulas, and supercongruences.

We prove rational alternating Ramanujan-type series of level $1$ discovered by the brothers David and Gregory Chudnovky, by using a method of the author. We have carried out the computations with Maple (a symbolic software for mathematics)., Comment: 6 pages

We use Zeilberger's algorithm for proving some identities of Ramanujan-type via $_2F_1$ evaluations., Comment: 8 pages + Appendix (new). In the Appendix we prove a rapid formula for the computation of the lemniscate constant

First we give formulas for proving 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 polynomials., Comment: 10 pages

Inspired by a Zudilin-Zhao's supercongruences pattern related to Ramanujan-like series for $1/\pi^k$, we conjecture a kind of $p$-adic expansions., Comment: 5 pages

We prove a kind of bilateral semi-terminating series related to Ramanujan-like series for negative powers of $\pi$, and conjecture a type of supercongruences associated to them. We support this conjecture by checking all the cases for many primes. In addition we are able to prove a few of them from some terminating hypergeometric identities. Finally, we make an intriguing observation., Comment: 18 pages

Using a modular equation of level $3$ and degree $23$ due to Chan and Liaw, we prove the fastest convergent rational Ramanujan-type series for $1/\pi$ of level $3$., Comment: 5 pages

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., Comment: 10 pages

We show with some examples how to prove some Ramanujan-type series for $1/\pi$ in an elementary way by using terminating identities., Comment: 4 pages

We prove q-analogues of two Ramanujan-type series for $1/\pi$ from $q$-analogues of ordinary WZ pairs., Comment: 5 + (3 appendix) pages. Extended version

We consider an extension of the Ramanujan series with a variable $x$. If we let $x=x_0$, we call the resulting series: "Ramanujan series with the shift $x_0$". Then, we relate these shifted series to some $q$-series and solve the case of level $4$ with the shift $x_0=1/2$. Finally, we indicate a possible way towards proving some patterns observed by the author corresponding to the levels $\ell=1, 2, 3$ and the shift $x_0=1/2$., Comment: 13 pages. Extended and improved version

We prove a formula for the Mangoldt function which relates it to a sum over all the non-trivial zeros of the Riemann zeta function, in addition we analize a truncated version of it., Comment: 15 pages. We have added a new formula for the spectrum of the prime numbers

Using a self-replicating method, we generalize with a free parameter some Borwein algorithms for the number $\pi$. This generalization includes values of the Gamma function like $\Gamma(1/3)$, $\Gamma(1/4)$ and of course $\Gamma(1/2)=\sqrt{\pi}$. In addition, we give new rapid algorithms for the perimeter of an ellipse., Comment: Some typos corrected (7 pages)

We show how to prove the examples of a paper by Chu and Zhang using the WZ-algorithm., Comment: Preprint 6 pages + Addendum 2 pages (The addendum is not in the journal reference)

We define bilateral series related to Ramanujan-like series for $1/\pi^2$. Then, we conjecture a property of them and give some applications., Comment: 13 pages

Special arithmetic series $f(x)=\sum_{n=0}^{\infty}c_nx^n$, whose coefficients $c_n$ are normally given as certain binomial sums, satisfy "self-replicating" functional identities. For example, the equation $$\frac1{(1+4z)^2}f\biggl(\frac{z}{(1+4z)^3}\biggr)=\frac1{(1+2z)^2}f\biggl(\frac{z^2}{(1+2z)^3}\biggr)$$ generates a modular form $f(x)$ of weight 2 and level 7, when a related modular parametrization $x=x(\tau)$ is properly chosen. In this note we investigate the potential of describing modular forms by such self-replicating equations as well as applications of the equations that do not make use of the modularity. In particular, we outline a new recipe of generating AGM-type algorithms for computing $\pi$ and other related constants. Finally, we indicate some possibilities to extend the functional equations to a two-variable setting., Comment: 16 pages

We use a method of translation to recover Borweins' quadratic and quartic iterations. Then, by using the WZ-method, we obtain some initial values which lead to the limit $1/\pi$. We will not use the modular theory nor either the Gauss' formula that we used in another paper. Our proofs are short and self-contained., Comment: 8 pages

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., Comment: Two typos in the formulas corrected. The addendum is not in the Journal reference. 9 pages, Integral Transforms and Special Functions (Published on line: March 19, 2015)

We explain an experimental method to find CY-type differential equations of order $3$ related to modular functions of genus zero. We introduce a similar class of Calabi-Yau differential equations of order $5$, show several examples and make a conjecture related to some geometric invariants. We finish the paper with a few examples of seven order., Comment: 21 pages

We prove some identities, which involve the non-trivial zeros of the Riemann zeta function. From them we derive some convergent asymptotic expansions related to the work by Cram\'er, and also new representations for some arithmetical functions in terms of the non-trivial zeros., Comment: Improved version. (17 pages)

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., Comment: 15 pages

We prove that there is a correspondence between Ramanujan-type formulas for 1/\pi, and formulas for Dirichlet L-values. The same method also allows us to resolve certain values of the Epstein zeta function in terms of rapidly converging hypergeometric functions. The Epstein zeta functions were previously studied by Glasser and Zucker., Comment: 28 pages

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.

Using the theory of Calabi-Yau differential equations we obtain all the parameters of Ramanujan-Sato-like series for $1/\pi^2$ as $q$-functions valid in the complex plane. Then we use these q-functions together with a conjecture to find new examples of series of non-hypergeometric type. To motivate our theory we begin with the simpler case of Ramanujan-Sato series for $1/\pi$., Comment: The Addendum is new

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 $1/\pi^2$., Comment: Identities for $1/pi^2$ are added

We prove, by the WZ-method, some hypergeometric identities which relate ten extended Ramanujan type series to simpler hypergeometric series. The identities we are going to prove are valid for all the values of a parameter $a$ when they are convergent. Sometimes, even if they do not converge, they are valid if we consider these identities as limits.

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

Using the machinery from the theory of Calabi-Yau differential equations, we find formulas for $1/\pi^2$ of hypergeometric and non-hypergeometric types., Comment: Some mistakes corrected and new formulas added

We generalize the patterns of supercongruences of Ramanujan-type observed by L. Van Hamme and W. Zudilin to series involving simple square roots anywhere and not only in the result of the sum. To support our observations we give some examples.

We use the Wilf-Zeilberger method to prove identities between Mahler measures of polynomials. In particular, we offer a new proof of a formula due to Lal\'{i}n, and we show how to translate the identity into a formula involving elliptic dilogarithms. This work settles a challenge problem proposed by Kontsevich and Zagier in their paper "Periods"., Comment: 15 pages

"Divergent" Ramanujan-type series for $1/\pi$ and $1/\pi^2$ provide us with new nice examples of supercongruences of the same kind as those related to the convergent cases. In this paper we manage to prove three of the supercongruences by means of the Wilf--Zeilberger algorithmic technique., Comment: 14 pages

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 paper we prove theorems related to the Ramanujan-type series for $1/\pi$ (type $_3F_2$) and to the Ramanujan-like series, discovered by the author, for $1/\pi^2$ (type $_5F_4$). Our developments for the cases $_3 F_2$ and $_5 F_4$ connect with the theory of modular functions and with the theory of Calabi-Yau differential equations, respectively.

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 certain series of this type., Comment: 12 pages (preprint) + 1 page (addendum). The addendum (A Maple program) is not in the Journal reference

Throughout more than two millennia many formulas have been obtained, some of them beautiful, to calculate the number pi. Among them, we can find series, infinite products, expansions as continued fractions and expansions using radicals. Some expressions which are (amazingly) related to pi have been evaluated. In addition, a continual battle has been waged just to break the records computing digits of this number; records have been set using rapidly converging series, ultra fast algorithms and really surprising ones, calculating isolated digits. The development of powerful computers has played a fundamental role in these achievements of calculus., Comment: Translated from the original Spanish by Luis Sanchez Lajusticia. Updated by the author. Some mistakes corrected

In 1987 Jonathan and Peter Borwein, inspired by the works of Ramanujan, derived many efficient algorithms for computing $\pi$. We will see that by using only a formula of Gauss's and elementary algebra we are able to prove the correctness of two of them., Comment: To appear in The American Mathematical Monthly. (6 pages)

The two-fold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for $\zeta(2)$ and $\zeta(3),$ and those of the second author for Euler's constant $\gamma$ and its alternating analog $\ln(4/\pi),$ and on the other hand the infinite products of the first author for $e$, and of the second author for $\pi$ and $e^\gamma.$ We obtain new double integral and infinite product representations of many classical constants, as well as a generalization to Lerch's transcendent of Hadjicostas's double integral formula for the Riemann zeta function, and logarithmic series for the digamma and Euler beta functions. The main tools are analytic continuations of Lerch's function, including Hasse's series. We also use Ramanujan's polylogarithm formula for the sum of a particular series involving harmonic numbers, and his relations between certain dilogarithm values., Comment: 21 pages, to appear in The Ramanujan Journal. Added Corollary 3.3, Lemma 3.1, Equations (5) and (33), all or part of Examples 3.5, 3.6, 3.7, 3.9, 3.11, 3.14, 5.12, and references [1], [16], [17], [18]. Omitted the old [4]. Modified Equation (49), Theorem 5.3, and Examples 3.25 and 3.26. Expanded the Abstract and Introduction. Rearranged Sections 2 and 3

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