# Rational analogues of Ramanujan's series for 1/π†

@article{Chan2012RationalAO, title={Rational analogues of Ramanujan's series for 1/$\pi$†}, author={Heng Huat Chan and Shaun Cooper}, journal={Mathematical Proceedings of the Cambridge Philosophical Society}, year={2012}, volume={153}, pages={361 - 383} }

Abstract A general theorem is stated that unifies 93 rational Ramanujan-type series for 1/π, 40 of which are believed to be new. Moreover, each series is shown to have a companion identity, thereby giving another 93 series, the majority of which are new.

#### 52 Citations

Proof of a rational Ramanujan-type series for 1/π. The fastest one in level 3

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- 2018

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 $3$.

Level 16 analogue of Ramanujan's theories of elliptic functions to alternative bases

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- 2016

Abstract The analogous theory for level 16 of Ramanujan's theories of elliptic functions to alternative bases is developed by studying the level 16 modular function h ( q ) = q ∏ j = 1 ∞ ( 1 − q 16 j… Expand

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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… Expand

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- 2015

We briefly review Ramanujan’s theories of elliptic functions to alternative bases, describe their analogues for levels 5 and 7, and develop new theories for levels 14 and 15. This gives rise to a… Expand

SOME NEW SERIES FOR 1 / π AND RELATED CONGRUENCES

- 2011

In this paper we prove some new series for 1/π as well as related congruences. We also raise several new kinds of series for 1/π and present related conjectural congruences involving representations… Expand

Holonomic alchemy and series for $1/\pi$

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- 2015

We adopt the "translation" as well as other techniques to express several identities conjectured by Z.-W. Sun in arXiv:1102.5649v47 by means of known formulas for $1/\pi$ involving Domb and other… Expand

Hypergeometric transformation formulas of degrees 3, 7, 11 and 23

- Mathematics
- 2015

Abstract The theory of theta functions is used to derive hypergeometric transformation formulas of degrees 3, 7, 11 and 23. As a consequence of the theory that is developed, some new series for 1 / π… Expand

Special Hypergeometric Motives and Their L-Functions: Asai Recognition

- Mathematics, Physics
- 2019

We recognize certain special hypergeometric motives, related to and inspired by the discoveries of Ramanujan more than a century ago, as arising from Asai $L$-functions of Hilbert modular forms.

Some new series for $1/\pi$ and related congruences

- Mathematics
- 2011

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… Expand

Lost in translation

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- 2012

The principal goal of this note is to set some grounds about applicability of algebraic transformations of (arithmetic) hypergeometric series for proving Ramanujan’s formulae for 1/π and their… Expand

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