Unification of the Cubic Analogues of the Jacobian Theta-Function

A common generalization a(q, ζ, z) of Hirschhorn-Garvan-Bonvein cubic analogues a(q, z), b(q, z), and c(q, z) of the classical theta-functions θ2(q, z), θ3(q, z), and θ4(q, z) is introduced. Some properties of a(q, ζ, z) are simply established and are shown to unify several known properties of a(q, z), b(q, z), and c(q, z).