Related papers: Ramanujan-type formulae for $1/\pi$: $q$-analogues
In terms of the hypergeometric method, we establish the extensions of two formulas for $1/\pi$ due to Ramanujan [27]. Further, other five summation formulas for $1/\pi$ with free parameters are also derived in the same way.
In this article we use theoretical and numerical methods to evaluate in a closed-exact form the parameters of Ramanujan type $1/\pi$ formulas.
We present several supercongruences that may be viewed as $p$-adic analogues of Ramanujan-type series for $1/\pi$ and $1/\pi^2$, and prove three of these examples.
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…
Using the machinery from the theory of Calabi-Yau differential equations, we find formulas for $1/\pi^2$ of hypergeometric and non-hypergeometric types.
In terms of the difference operators, we establish several curious transformation and summation formulas for basic hypergeometric series. When the parameters are specified, they produce $q$-analogues of Ramanujan's three series for 1/$\pi$…
In 1997, Van Hamme proposed 13 supercongruences corresponding to $1/\pi$ series of the Ramanujan-type. Inspired by the recent work of V.J.W. Guo, we establish a unified $q$-analogue of Van Hamme's (B.2), (E.2) and (F.2) supercongruences,…
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.
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…
In 1914, Ramanujan gave a list of 17 identities expressing $1/\pi$ as linear combinations of values of hypergeometric functions at certain rational numbers. Since then, identities of similar nature have been discovered by many authors.…
In this article we give the theoretical background for generating Ramanujan type $1/\pi^{2\nu}$ formulas. As applications of our method we give a general construction of $1/\pi^4$ series and examples of $1/\pi^6$ series. We also study the…
In 1997, van Hamme developed $p-$adic analogs, for primes $p$, of several series which relate hypergeometric series to values of the gamma function, originally studied by Ramanujan. These analogs relate truncated sums of hypergeometric…
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.
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.
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…
Inspired by the recent work on $q$-congruences and the quadratic summation formula of Rahman, we provide some new $q$-supercongruences. By taking $q\to 1$ in one of our results, we obtain a new Ramanujan-type supercongruence, which has the…
Using some properties of the gamma function and the well-known Gauss summation formula for the classical hypergeometric series, we prove a four-parameter series expansion formula, which can produce infinitely many Ramanujan type series for…
We explain the use and set grounds about applicability of algebraic transformations of arithmetic hypergeometric series for proving Ramanujan's formulae for $1/\pi$ and their generalisations.
Several terminating generalizations of Ramanujan's formula for $\frac{1}{\pi}$ with complete WZ proofs are given.