Cubic theta functions

Some new identities for the four cubic theta functions a′(q,z), a(q,z), b(q,z) and c(q,z) are given. For example, we show thata′(q,z)3=b(q,z)3+c(q)2c(q,z).This is a counterpart of the identitya(q,z)3=b(q)2b(q,z3)+c(q,z)3,which was found by Hirschhorn et al. urent series expansions of the four cubic theta functions are given. Their transformation properties are established using an elementary approach due to K. Venkatachaliengar. By applying the modular transformation to the identities given by Hirschhorn et al., several new identities in which a′(q,z) plays the role of a(q,z) are obtained.