East China Normal University

The main content of this paper is listed in the following.Ⅰ. We prove a general alternate circular summation formula of theta func-tions, which implies a great deal of theta-function identities. In particular, we recover several identities in Ramanujan's Notebook from this identity. By using the Jacobi imaginary transformation to the alternate circular summation for-mula, we get its dual forms. Some special cases of the dual forms are discussed. We get four formulaes for (q;q)∞2n.Ⅱ. We give very natural proofs of the quintuple product identity, the Mac-donald identity for A2 and the Winquist identity. Our tools are the Jacobi triple product identity and the approach of series rearrangement. We also establish some series-product identities, three of which simplifies the corresponding iden-tities in one of the references. A great deal of modular identities, including identities for integer powers of Dedekind eta functionη(τ), are obtained.Ⅲ. We get a new identity for the product of four theta functions and an identity for(q;q)∞12.Ⅳ. We obtain the solutions of four q-functional equations and express the so-lutions in q-operator forms. These equations give sufficient conditions for Cauchy operator methods. We also give simple proofs of the bilateral series 2ψ2,3ψ3 and 4ψ4, respectively, using the method of analytic continuation.