Corpus ID: 117829947

# Ramanujan-type formulae for $1/\pi$: The art of translation

@article{Guillera2013RamanujantypeFF,
title={Ramanujan-type formulae for \$1/\pi\$: The art of translation},
author={Jes{\'u}s Guillera and Wadim Zudilin},
journal={arXiv: Number Theory},
year={2013}
}
• Published 3 February 2013
• Mathematics
• arXiv: Number Theory
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.
16 Citations
A method for proving Ramanujan’s series for $$1/\pi$$1/π
In a famous paper of 1914 Ramanujan gave a list of 17 extraordinary formulas for the number $$1/\pi$$1/π. In this paper we explain a general method to prove them, based on some ideas of James WanExpand
Proofs of some Ramanujan series for $1/\pi$ using a Zeilberger's program
We show with some examples how to prove some Ramanujan-type series for $1/\pi$ in an elementary way by using terminating identities.
Holonomic alchemy and series for $1/\pi$
• Mathematics
• 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 otherExpand
WZ proofs of Ramanujan-type series (via $_2F_1$ evaluations)
We use Zeilberger's algorithm for proving some identities of Ramanujan-type via $_2F_1$ evaluations.
A family of Ramanujan–Orr formulas for 1/π
We use a variant of Wan's method to prove two Ramanujan–Orr type formulas for . This variant needs to know in advance the formulas for that we want to prove, but avoids the need of solving a systemExpand
Proofs of some Ramanujan series for 1/π using a program due to Zeilberger
• Jesús Guillera
• Mathematics
• Journal of Difference Equations and Applications
• 2018
ABSTRACT We show with some examples how to prove some Ramanujan-type series for in an elementary way by using terminating identities.
Level 14 and 15 analogues of Ramanujan’s elliptic functions to alternative bases
• Mathematics
• 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 aExpand
About a class of Calabi-Yau differential equations
• Mathematics
• 2013
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 ofExpand
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.
HYPERGEOMETRIC MODULAR EQUATIONS
• Mathematics
• Journal of the Australian Mathematical Society
• 2018
We record $\binom{42}{2}+\binom{23}{2}+\binom{13}{2}=1192$ functional identities that, apart from being amazingly amusing in themselves, find application in the derivation of Ramanujan-type formulasExpand

#### References

SHOWING 1-10 OF 48 REFERENCES
Extensions of Ramanujan's two formulas for $1/\pi$
• Mathematics
• 2012
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 alsoExpand
Rational analogues of Ramanujan's series for 1/π†
• Mathematics
• Mathematical Proceedings of the Cambridge Philosophical Society
• 2012
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, therebyExpand
Ramanujan-type formulae for 1/π: a second wind?
In 1914 S. Ramanujan recorded a list of 17 series for 1=…. We survey the methods of proofs of Ramanujan’s formulae and indicate recently discovered generalisations, some of which are not yet proven.Expand
Ramanujan-Sato-Like Series
• Mathematics, Computer Science
• Number Theory and Related Fields
• 2013
Using the theory of Calabi–Yau differential equations, all the parameters of Ramanujan–Sato-like series for 1∕π 2 are obtained as q-functions valid in the complex plane and used to find new examples of series of non-hypergeometric type. Expand
Divergent Ramanujan-type supercongruences
• Mathematics
• 2012
"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 toExpand
Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity
• Mathematics
• 1988
Complete Elliptic Integrals and the Arithmetic-Geometric Mean Iteration. Theta Functions and the Arithmetic-Geometric Mean Iteration. Jacobi's Triple-Product and Some Number Theoretic Applications.Expand
On WZ-pairs which prove Ramanujan series
The known WZ-proofs for Ramanujan-type series related to 1/π gave us the insight to develop a new proof strategy based on the WZ-method. Using this approach we are able to find more generalizationsExpand
Hypergeometric evaluation identities and supercongruences
We apply some hypergeometric evaluation identities, including a strange valuation of Gosper, to prove several supercongruences related to special valuations of truncated hypergeometric series. InExpand
Algebraic Transformations of Gauss Hypergeometric Functions
This article gives a classification scheme of algebraic transformations of Gauss hypergeometric functions, or pull-back transformations between hypergeometric differential equations. TheExpand
New representations for Apéry-like sequences
• Mathematics
• 2010
We prove algebraic transformations for the generating series of three Apery-like sequences. As application, we provide new binomial representations for the sequences. We also illustrate a method thatExpand