On the theory of elliptic functions based on ₂𝐹₁(\frac{1}3,\frac{2}3;\frac{1}2;𝑧)

Shen, Li Chien

doi:10.1090/s0002-9947-04-03600-1

Trans. Amer. Math. Soc.

2004

DOI: 10.1090/s0002-9947-04-03600-1

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On the theory of elliptic functions based on ₂𝐹₁(\frac{1}3,\frac{2}3;\frac{1}2;𝑧)

Li Chien Shen

Abstract: Abstract. Based on properties of the hypergeometric series 2 F 1 ( ; z), we develop a theory of elliptic functions that shares many striking similarities with the classical Jacobian elliptic functions.

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Cited by 5 publications

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Quantum geometry of resurgent perturbative/nonperturbative relations

Başar

Dunne

Ünsal

2017

J. High Energ. Phys.

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For a wide variety of quantum potentials, including the textbook 'instanton' examples of the periodic cosine and symmetric double-well potentials, the perturbative data coming from fluctuations about the vacuum saddle encodes all non-perturbative data in all higher non-perturbative sectors. Here we unify these examples in geometric terms, arguing that the all-orders quantum action determines the all-orders quantum dual action for quantum spectral problems associated with a classical genus one elliptic curve. Furthermore, for a special class of genus one potentials this relation is particularly simple: this class includes the cubic oscillator, symmetric double-well, symmetric degenerate triple-well, and periodic cosine potential. These are related to the Chebyshev potentials, which are in turn related to certain N = 2 supersymmetric quantum field theories, to mirror maps for hypersurfaces in projective spaces, and also to topological c = 3 Landau-Ginzburg models and 'special geometry'. These systems inherit a natural modular structure corresponding to Ramanujan's theory of elliptic functions in alternative bases, which is especially important for the quantization. Insights from supersymmetric quantum field theory suggest similar structures for more complicated potentials, corresponding to higher genus. Our approach is very elementary, using basic classical geometry combined with all-orders WKB.

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Quantum geometry of resurgent perturbative/nonperturbative relations

Başar

Dunne

Ünsal

2017

J. High Energ. Phys.

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On Hecke groups, Schwarzian triangle functions and a class of hyper-elliptic functions

Lei

2016

Ramanujan J

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Let m be a positive integer ≥3 and λ = 2 cos π m . The Hecke group G(λ) is generated by the fractional linear transformations τ + λ and − 1 τ for τ in the upper half plane H of the complex plane C. We consider a set of functions f 0 , f i and f ∞ automorphic with respect to G(λ), constructed from the conformal mapping of the fundamental domain of G(λ) to the upper half plane H, and establish their connection with the Legendre functions and a class of hyper-elliptic functions. Many well-known classical identities associated with the cases of λ = 1 and 2 are preserved. As an application, we will establish a set of identities expressing the reciprocal of π in terms of the hypergeometric series.

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An addition formula for the Jacobian theta function and its applications

Liu

2007

Advances in Mathematics

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On the theory of elliptic functions based on ₂𝐹₁(\frac{1}3,\frac{2}3;\frac{1}2;𝑧)

Abstract: Abstract. Based on properties of the hypergeometric series 2 F 1 ( ; z), we develop a theory of elliptic functions that shares many striking similarities with the classical Jacobian elliptic functions.

Cited by 5 publications

References 5 publications

Quantum geometry of resurgent perturbative/nonperturbative relations

Quantum geometry of resurgent perturbative/nonperturbative relations

On Hecke groups, Schwarzian triangle functions and a class of hyper-elliptic functions

An addition formula for the Jacobian theta function and its applications

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