In this paper, we give some extensions for Ramanujan’s circular summation formulas with the mixed products of two Jacobi’s theta functions. As applications, we also obtain many interesting identities of Jacobi’s theta functions.

中文翻译：

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Ramanujan循环求和公式和应用程序的某些扩展

在本文中，我们用两个Jacobi theta函数的混合乘积来扩展Ramanujan的循环求和公式。作为应用程序，我们还获得了Jacobi theta函数的许多有趣的恒等式。