# Sporadic sequences, modular forms and new series for 1/π

@article{Cooper2012SporadicSM, title={Sporadic sequences, modular forms and new series for 1/$\pi$}, author={Shaun Cooper}, journal={The Ramanujan Journal}, year={2012}, volume={29}, pages={163-183} }

Two new sequences, which are analogues of six sporadic examples of D. Zagier, are presented. The connection with modular forms is established and some new series for 1/π are deduced. The experimental procedure that led to the discovery of these results is recounted. Proofs of the main identities will be given, and some congruence properties that appear to be satisfied by the sequences will be stated as conjectures.

#### 36 Citations

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

Sequences, modular forms and cellular integrals

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- Mathematical Proceedings of the Cambridge Philosophical Society
- 2018

Abstract It is well-known that the Apéry sequences which arise in the irrationality proofs for ζ(2) and ζ(3) satisfy many intriguing arithmetic properties and are related to the pth Fourier… Expand

New Representations for all Sporadic Apéry-Like Sequences, With Applications to Congruences

- Mathematics
- Experimental Mathematics
- 2021

Sporadic Apéry-like sequences were discovered by Zagier, by Almkvist and Zudilin and by Cooper in their searches for integral solutions for certain families of secondand third-order differential… Expand

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

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- 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, thereby… Expand

Interpolated Sequences and Critical L-Values of Modular Forms

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- Texts & Monographs in Symbolic Computation
- 2019

Recently, Zagier expressed an interpolated version of the Apery numbers for \(\zeta (3)\) in terms of a critical L-value of a modular form of weight 4. We extend this evaluation in two directions. We… Expand

A MODULAR SUPERCONGRUENCE FOR 6F5: AN APÉRY-LIKE STORY

- 2017

Abstract. We prove a supercongruence modulo p between the pth Fourier coefficient of a weight 6 modular form and a truncated 6F5-hypergeometric series. Novel ingredients in the proof are the… Expand

Generalized Lucas congruences and linear $p$-schemes

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- 2021

We observe that a sequence satisfies Lucas congruences modulo p if and only if its values modulo p can be described by a linear p-scheme, as introduced by Rowland and Zeilberger, with a single state.… Expand

q-Congruences, with applications to supercongruences and the cyclic sieving phenomenon

- Mathematics
- International Journal of Number Theory
- 2019

We establish a supercongruence conjectured by Almkvist and Zudilin, by proving a corresponding [Formula: see text]-supercongruence. Similar [Formula: see text]-supercongruences are established for… Expand

Divisibility properties of sporadic Apéry-like numbers

- Mathematics
- 2015

In 1982, Gessel showed that the Apéry numbers associated to the irrationality of ζ(3) satisfy Lucas congruences. Our main result is to prove corresponding congruences for all known sporadic… Expand

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 a… Expand

#### References

SHOWING 1-10 OF 24 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's cubic continued fraction revisited

- Mathematics
- 2007

In this article, we derive a sequence of numbers which converge to 1/π. We will also derive a new series for 1/π. These new results are motivated by the study of Ramanujan’s cubic continued fraction.

New analogues of Clausen’s identities arising from the theory of modular forms | NOVA. The University of Newcastle's Digital Repository

- Mathematics, Engineering
- 2011

Abstract Around 1828, T. Clausen discovered that the square of certain hypergeometric F 1 2 function can be expressed as a hypergeometric F 2 3 function. Special cases of Clausenʼs identities were… Expand

Ramanujan-type formulae for 1/pi: a second wind?

- Mathematics
- 2008

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

Quintic and septic Eisenstein series

- Mathematics
- 2009

Using results that were well-known to Ramanujan, we give proofs of some results for Eisenstein series in the lost notebook. Our proofs have the additional advantage that it is not necessary to know… Expand

Generalizations of Clausen's Formula and algebraic transformations of Calabi–Yau differential equations

- Mathematics
- Proceedings of the Edinburgh Mathematical Society
- 2011

Abstract We provide certain unusual generalizations of Clausen's and Orr's theorems for solutions of fourth-order and fifth-order generalized hypergeometric equations. As an application, we present… Expand

New 5F4 hypergeometric transformations, three-variable Mahler measures, and formulas for 1/π

- Mathematics
- 2007

Abstract
New relations are established between families of three-variable Mahler measures. Those identities are then expressed as transformations for the 5F4 hypergeometric function. We use these… Expand

On Rationally Parametrized Modular Equations

- Mathematics
- 2006

Many rationally parametrized elliptic modular equations are derived. Each comes from a family of elliptic curves attached to a genus-zero congruence subgroup $\Gamma_0(N)$, as an algebraic… Expand

Modular Forms and String Duality

- Mathematics
- 2008

Aspects of arithmetic and modular forms: Motives and mirror symmetry for Calabi-Yau orbifolds by S. Kadir and N. Yui String modular motives of mirrors of rigid Calabi-Yau varieties by S. Kharel, M.… Expand

The Apéry numbers, the Almkvist-Zudilin numbers and new series for 1/\pi

- Mathematics
- 2009

This paper concerns series for $1/\pi$, such as Sato's series (\ref{eqn:sato}), and the series of H.H. Chan, S.H. Chan and Z.-G. Liu (\ref{eqn:exampleforb_ks}) below.The examples of Sato, Chan, Chan… Expand