Analogues of Jacobi’s derivative formula
Analogues of Jacobi’s derivative formula
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雅可比导数公式的类似物
DOI：
10.1007/s11139-015-9715-7
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
K. Matsuda
中科院分区：
文献类型：
--
作者：
K. Matsuda
In this paper, we obtain analogues of Jacobi’s derivative formula in terms of the theta constants with rational characteristics. For this purpose, we use the arithmetic formulas of the number of representations of a natural numbernas the sum of two squares, or the sum of a square and twice a square, which is given by $$\begin{aligned} S_2(n)&=\sharp \{(x,y)\in \mathbb {Z}^2 \,| \, n=x^2+y^2 \} =4\sum _{d|n, \,d{:}\mathrm{odd}} (-1)^{\frac{d-1}{2}}, \\ S_{1,2}(n)&=\sharp \{(x,y)\in \mathbb {Z}^2 \,| \, n=x^2+2y^2 \} =2(d_{1,8}(n)+d_{3,8}(n)-d_{5,8}(n)-d_{7,8}(n)), \end{aligned}$$where for the positive integersj,k,n,denotes the number of positive divisorsdofnsuch that.
In this paper, we obtain analogues of Jacobi’s derivative formula in terms of the theta constants with rational characteristics. For this purpose, we use the arithmetic formulas of the number of representations of a natural numbernas the sum of two squares, or the sum of a square and twice a square, which is given by $$\begin{aligned} S_2(n)&=\sharp \{(x,y)\in \mathbb {Z}^2 \,| \, n=x^2+y^2 \} =4\sum _{d|n, \,d{:}\mathrm{odd}} (-1)^{\frac{d-1}{2}}, \\ S_{1,2}(n)&=\sharp \{(x,y)\in \mathbb {Z}^2 \,| \, n=x^2+2y^2 \} =2(d_{1,8}(n)+d_{3,8}(n)-d_{5,8}(n)-d_{7,8}(n)), \end{aligned}$$where for the positive integersj,k,n,denotes the number of positive divisorsdofnsuch that.