标题： 扩展的兰登公式和双重乘积雅可比θ函数 显示英文标题

标题： Extended Landen's formulas and double product theta functions

摘要： 对于任意正整数$p$，Landen 公式被扩展，以用模$p\tau$的 theta 函数表示由模$\tau$的 theta 函数的$p$乘积，该方法应用于几个示例。 接下来表明， genus$g=1$的 theta 函数的双重乘积可以写成$g=2$个 theta 函数的和，这是具有特殊周期矩阵的$\tau_{11}=\tau_{22}$的一个子集。 展示了几个应用示例，其中包括 Borwein 和 Borwein 的三次恒等式。 显示英文摘要

摘要： For an arbitrary positive integer $p$, Landen's formula is extended to express theta function with modulus $p\tau$ by $p$ product of theta functions with $\tau$, which is applied to several examples. Next it is shown that double product of theta functions of genus $g=1$ is written by a sum of $g=2$ theta functions, which is a subset having a special period matrix of $\tau_{11}=\tau_{22}$. Several applied examples are shown, which include the cubic identity of Borwein and Borwein.