Eisenstein series associated with Γ0(2)

A set of nonlinear differential equations associated with the Eisenstein series of the congruent subgroup Γ 0 (2) of the modular group SL 2 (Z) is constructed. These nonlinear equations are analogues of the well known Ramanujan equations, as well as the Chazy and Darboux-Halphen equations associated with the modular group. The general solutions of these equations can be realized in terms of the Schwarz triangle function S(0, 0, 1/2; z).

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“…(1.7) In the same manner as (1.6), these functions P(q), P(q) and Q(q) satisfy the differential equations [17,18,19] …”

Section: Introductionmentioning

confidence: 99%

Integrable systems and modular forms of level 2

Ablowitz¹,

Chakravarty²,

Hahn³

2006

J. Phys. A: Math. Gen.

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“…(3.5c) Identity (3.5a) was observed by Hahn [13] while (3.5b) and (3.5c) were recorded by Huber [16].…”

Section: Corollary 32mentioning

confidence: 78%

“…She (see also [13]) used these to derive two families of differential equations involving Eisenstein series, one of which is the following…”

Section: Introductionmentioning

confidence: 99%

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Differential equations satisfied by Eisenstein series of level 2

Toh

2011

Ramanujan J

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Ramanujan's differential equations for the classical Eisenstein series are of great importance to many areas in number theory and special functions. H.H. Chan recently demonstrated that these differential equations can be derived from the triple product identity and the quintuple product identity in an elementary manner. In this article, we extend this method in a uniform manner to derive corresponding differential equations for the Eisenstein series of level 2. Several applications of these differential equations are also given.

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“…Ramamani [30,31] (see also [18]) utilized Ramanujan's results and an additional trigonometric series identity to show that the related series e(q) := 1 + 24 ∞ n=1 nq n 1 + q n , P(q) :…”

Section: Introductionmentioning

confidence: 99%

Basic representations for Eisenstein series from their differential equations

Huber

2009

Journal of Mathematical Analysis and Applications

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Complete elliptic integrals Ramanujan's alternative theories of elliptic functions Modular J -function Lie symmetryIn this paper we provide a new approach for the derivation of parameterizations for the Eisenstein series. We demonstrate that a variety of classical formulas may be derived in an elementary way, without knowledge of the inversion formulae for the corresponding Schwarzian triangle functions. In particular, we provide a new derivation for the parametric representations of the Eisenstein series in terms of the complete elliptic integral of the first kind. The proof given here is distinguished from existing elementary proofs in that we do not employ the Jacobi-Ramanujan inversion formula relating theta functions and hypergeometric series. Our alternative approach is based on a Lie symmetry group for the differential equations satisfied by certain Eisenstein series. We employ similar arguments to obtain parameterizations from Ramanujan's alternative signatures and those associated with the inversion formula for the modular J -function. Moreover, we show that these parameterizations represent the only possible signatures under a certain assumed form for the Lie group parameters.

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Eisenstein series associated with Γ0(2)

Cited by 13 publications

References 13 publications

Integrable systems and modular forms of level 2

Integrable systems and modular forms of level 2

Differential equations satisfied by Eisenstein series of level 2

Basic representations for Eisenstein series from their differential equations

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