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AN ALTERNATE CIRCULAR SUMMATION FORMULA OF THETA FUNCTIONS AND ITS APPLICATIONSKeywords: Theta function , circular summation , elliptic function , Ramanujan , modular equation Abstract: We prove a general alternate circular summation formula of theta functions, which yields a great deal of theta-function identities. In particular, we recover several identities in Ramanujan's Notebook from this identity. We also obtain two formulas for $(q;q)_infty^{2n}$.
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