The Ramanujan Journal

, Volume 11, Issue 3, pp 355–397

Cubic elliptic functions

Article

Abstract

The function
$$\displaylines{\displaystyle \Phi(\theta;q) = \theta + 3\sum_{k=1}^\infty\frac{\sin(2k\theta)q^k}{k(1+q^k+q^{2k})}}$$
occurs in one of Ramanujan’s inversion formulas for elliptic integrals. In this article, a common generalization of the cubic elliptic functions
$$\displaylines{g_1(\theta;q) = \frac{1}{6} + \sum_{k=1}^\infty \frac{q^k}{1+q^k+q^{2k}}\cos k\theta,\cr g_2(\theta;q) = \frac{1}{2} \frac{\sin \frac{\theta}{2}}{\sin \frac{3\theta}{2}} + \sum_{k=1}^\infty \frac{\chi_3(k)q^k}{1-q^k} \cos k\theta,}$$
is given. The function g1 is the derivative of Ramanujan’s function Φ (after rescaling), and χ3(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 g1 and g2, are proved.

Keywords

Eisenstein series Elliptic function Modular transformation Ramanujan Theta function Venkatachaliengar

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