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

  title={Sporadic sequences, modular forms and new series for 1/$\pi$},
  author={Shaun Cooper},
  journal={The Ramanujan Journal},
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. 
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 representationsExpand
Sequences, modular forms and cellular integrals
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 FourierExpand
Rational analogues of Ramanujan's series for 1/π†
  • H. Chan, Shaun Cooper
  • 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
Interpolated Sequences and Critical L -Values of Modular Forms
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. WeExpand
New representations for all sporadic Ap\'ery-like sequences, with applications to congruences
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 differentialExpand
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 theExpand
q-Congruences, with applications to supercongruences and the cyclic sieving phenomenon
  • O. Gorodetsky
  • 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 forExpand
Divisibility properties of sporadic Apéry-like numbers
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 sporadicExpand
Level 14 and 15 analogues of Ramanujan’s elliptic functions to alternative bases
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
Calabi-Yau operators of degree two
We show that the solutions to the equations defining the so-called Calabi-Yau condition for fourth order operators of degree two defines a variety that consists of ten irreducible components. TheseExpand


Domb's numbers and Ramanujan–Sato type series for 1/π
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 πExpand
Ramanujan's cubic continued fraction revisited
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
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 wereExpand
Ramanujan-type formulae for 1/pi: 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
Quintic and septic Eisenstein series
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 knowExpand
Generalizations of Clausen's Formula and algebraic transformations of Calabi–Yau differential equations
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 presentExpand
New 5F4 hypergeometric transformations, three-variable Mahler measures, and formulas for 1/π
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 theseExpand
On Rationally Parametrized Modular Equations
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 algebraicExpand
Modular Forms and String Duality
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
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, ChanExpand