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An alternate circular summation formula of theta functions and its applications

Zhu Jun-Ming (Department of Mathematics, Luoyang Normal University, Luoyang City, Henan Province, China + Department of Mathematics, East China Normal University, Shanghai City, China)

We prove a general alternate circular summation formula of theta functions, which implies a great deal of theta-function identities. In particular, we recover several identities in Ramanujan's Notebook from this identity. We also obtain two formulaes for (q; q)2n∞.

Keywords: theta function, circular summation, elliptic function, Ramanujan, modular equation