A Cubic Counterpart of Jacobi's Identity and the AGM

Borwein, Jonathan M.; Borwein, Peter

doi:10.2307/2001551

Cited by 76 publications

(98 citation statements)

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“…We begin by recalling the Lambert series for a(q) and the product representations of b(q) and c(q) [1]: 3 1 − q n .…”

Section: Proofs Of Identities Involving A(q) B(q) C(q) and The Weiementioning

confidence: 99%

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On the points of order 3 of the period parallelogram of the Weierstrass elliptic function

Lei

2008

Ramanujan J

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“…We begin by recalling the Lambert series for a(q) and the product representations of b(q) and c(q) [1]: 3 1 − q n .…”

Section: Proofs Of Identities Involving A(q) B(q) C(q) and The Weiementioning

confidence: 99%

“…The original proof of this identity based on a combinatorial approach is given by Borwein and Borwein [1] and Borwein, Borwein and Garvan in [2]. In [5], Liu provided an alternative proof using the properties of the theta functions.…”

Section: From Lemmas 32 and 33 We Derivementioning

confidence: 99%

On the points of order 3 of the period parallelogram of the Weierstrass elliptic function

Lei

2008

Ramanujan J

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“…We also record, for future reference, the properties [5], [6] and [16]. 1 The function c(q, z) in [13] differs from the one defined here by a factor of q …”

Section: The Cubic Theta Functionsmentioning

confidence: 99%

“…Equation(11.8) was given by Berndt et al[3, p. 108],[4, Corollary 4.6]. Equations (11.6) and (11.7) were given without proof by J. M. and P. B. Borwein[5, Remark 2.4(iii)]. All of (11.6)-(11.8) were given by Liu[16, Eqs.…”

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confidence: 99%

Cubic elliptic functions

Cooper

2006

Ramanujan J

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The functionoccurs in one of Ramanujan's inversion formulas for elliptic integrals. In this article, a common generalization of the cubic elliptic functionsis given. The function g 1 is the derivative of Ramanujan's function (after rescaling), and χ 3 (n) = 0, 1 or −1 according as n ≡ 0, 1 or 2 (mod 3), respectively, and |q| < 1. Many properties of the common generalization, as well as the functions g 1 and g 2 , are proved.

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“…The referee has kindly pointed out that the expression for the function g(x) we shall give is not new but already appears in [1] and [4]. In a forthcoming publication together with Y. Godin we shall reprove that…”

Section: Final Remarksmentioning

confidence: 99%