# Ramanujan-Sato-Like Series

@inproceedings{Almkvist2013RamanujanSatoLikeS, title={Ramanujan-Sato-Like Series}, author={Gert Almkvist and Jes{\'u}s Guillera}, booktitle={Number Theory and Related Fields}, year={2013} }

Using the theory of Calabi–Yau differential equations we obtain all the parameters of Ramanujan–Sato-like series for 1∕π 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∕π.

#### 9 Citations

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

- Mathematics
- 2013

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

Kind of proofs of Ramanujan-like series

- Mathematics
- 2012

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… Expand

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 of… Expand

Rational Hypergeometric Ramanujan Identities for $1/\pi^c$: Survey and Generalizations

- Mathematics
- 2021

We give a simple unified proof for all existing rational hypergeometric ramanujan identities for 1/π, and give a complete survey (without proof) of several generalizations: rational hypergeometric… Expand

Arithmetic hypergeometric series

- Mathematics
- 2011

The main goal of this survey is to give common characteristics of auxiliary hypergeometric functions (and their generalisations), functions which occur in number-theoretic problems. Originally… Expand

Cutting and gluing with running couplings in $\mathcal{N}=2$ QCD

- Physics, Mathematics
- 2021

We consider the order parameter u = 〈Trφ2〉 as function of the running coupling constant τ ∈ H of asymptotically free N = 2 QCD with gauge group SU(2) and Nf ≤ 3 massive hypermultiplets. If the domain… Expand

Cutting and gluing with running couplings in N = 2 QCD

- 2021

We consider the order parameter u = 〈Trφ2〉 as function of the running coupling constant τ ∈ H of asymptotically free N = 2 QCD with gauge group SU(2) and Nf ≤ 3 massive hypermultiplets. If the domain… Expand

Výpočet Ludolfova čísla a ověření jeho vlastností

- Mathematics
- 2020

This thesis deals with analysis of available algorithms for calculating exact value of Ludolph’s number π. This thesis also describes geometric meaning, properties and brief history of Ludolph’s… Expand

Ramanujan-type $1/\pi$-series from bimodular forms

- Mathematics
- 2020

We develop an approach to establish $1/\pi$-series from bimodular forms. Utilizing this approach, we obtain new families of $2$-variable $1/\pi$-series associated to Zagier's sporadic Apery-like… Expand

#### References

SHOWING 1-10 OF 34 REFERENCES

Domb's numbers and Ramanujan–Sato type series for 1/π

- Mathematics
- 2004

Abstract In this article, we construct a general series for 1 π . We indicate that Ramanujan's 1 π -series are all special cases of this general series and we end the paper with a new class of 1 π… Expand

Ramanujan-like Series for 1/π2 and String Theory

- Computer Science, Mathematics
- Exp. Math.
- 2012

Using the machinery from the theory of Calabi–Yau differential equations, we find formulas for 1/π2 of hypergeometric and nonhypergeometric types.

Ramanujan-like series for $1/\pi^2$ and String Theory

- Mathematics
- 2010

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

A Matrix form of Ramanujan-type series for $1/\pi$

- Mathematics
- 2009

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… 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 to… Expand

The art of finding Calabi-Yau differential equations Dedicated to the 90-th birthday of Lars Garding.

- Mathematics
- 2010

In this paper various methods for finding Calabi-Yau differential equations are discussed. They are formalized versions of the differential equations satisfied by the periods of Calabi-Yau manifolds… Expand

Mosaic Supercongruences of Ramanujan Type

- Computer Science, Mathematics
- Exp. Math.
- 2012

Congruences of supercongruences of Ramanujan type observed by L. Van Hamme and W. Zudilin are presented, inspired by Ramanuja-type series that involve quadratic algebraic numbers. Expand

Complex series for 1/π

- Mathematics
- 2012

Many series for 1/π were discovered since the appearance of S. Ramanujan’s famous paper “Modular equations and approximation to π” published in 1914. Almost all these series involve only real… Expand

Ramanujan-type supercongruences

- Mathematics
- 2009

Abstract We present several supercongruences that may be viewed as p-adic analogues of Ramanujan-type series for 1 / π and 1 / π 2 , and prove three of these examples.

On Sp_4 modularity of Picard--Fuchs differential equations for Calabi--Yau threefolds (with an appendix by Vicentiu Pasol)

- Mathematics
- 2008

Motivated by the relationship of classical modular functions and Picard--Fuchs linear differential equations of order 2 and 3, we present an analogous concept for equations of order 4 and 5.