Cubic analogues of the Jacobian theta function (z,q)

Abstract: There are three modular forms $a(q)$, $b(q)$, $c(q)$ involved in the parametrization of the hypergeometric function ${}_2F_1(^{1\over 3},{{2\over 3}\atop 1};\cdot)$ analogous to the classical $\theta_2(q)$, $\theta_3(q)$, $\theta_4(q)$ and the hypergeometric function ${}_2F_1(^{1\over 2},{{1\over 2}\atop 1};\cdot)$. We give elliptic function generalizations of $a(q)$, $b(q)$, $c(q)$ analogous to the classical theta-function $\theta(z,q)$. A number of identities are proved. The proofs are self-contained, relying on nothing more than the Jacobi triple product identity.