Cubic elliptic functions (Q850539)

The cubic elliptic functions referred to in the title of this paper are ones which are analogous to \[ g_1(\theta,q)= \frac 16 + \sum_{1\leq k < \infty} \frac{q^{k}}{1+q^{k}+q^{2k}} \sin(2k \theta) \] which is an elliptic function in \(\theta\) with fundamental periods \(2 \pi\) and \(6\pi \imath t\) where \(q=e^{-2\pi t}\). The development given here is in the style of Jacobi and Ramanujan. The fundamental formulae are proved either by algebraic transformations or by complex function theory. An analogue of a formula of Venkatachaliengar plays a particularly important role in the development. The author proves a large number of formulae relating different functions in the class considered.