A bivariate circular summation formula for cubic theta functions and its implications

In this paper, a general bivariate circular summation formula and its dual form are established. They connect Ramanujan's theta function with the cubic analogue, Sigma(infinity)(m,n=-infinity) q(m2+mn+n2)x(m), of the classical theta functions introduced by M. Hirschhorn, F. Garvan and J. Borwein [18]. As applications, many new theta function identities are found and some well-known results are recovered from this summation formula as well as its dual form.