ヤコビの三重積
c ( τ , v ) = θ ( v , τ ) Θ ( v , τ ) = ∑ n = − ∞ ∞ e π i τ n 2 + 2 π i n v ∏ m = 1 ∞ ( 1 − e 2 m π i τ ) ( 1 + e ( 2 m − 1 ) π i τ + 2 π i v ) ( 1 + e ( 2 m − 1 ) π i τ − 2 π i v ) {\displaystyle c(\tau ,v)={\frac {\vartheta (v,\tau )}{\Theta (v,\tau )}}={\frac {\sum _{n=-\infty }^{\infty }{e^{{\pi }i{\tau }n^{2}+2{\pi }inv}}}{\prod _{m=1}^{\infty }{\left(1-e^{2m{\pi }i{\tau }}\right)\left(1+e^{(2m-1){\pi }i{\tau }+2{\pi }iv}\right)\left(1+e^{(2m-1){\pi }i{\tau }-2{\pi }iv}\right)}}}}
は v {\displaystyle v} に依存しない。
S ( x , z ) = S ( x , z x ) 1 − x z = 1 1 − x z ( 1 + b 1 ( x ) x z 1 − x 2 z + b 2 ( x ) x 2 z 2 ( 1 − x 2 z ) ( 1 − x 3 z ) + ⋯ ) = 1 1 − x z + b 1 ( x ) x z ( 1 − x z ) ( 1 − x 2 z ) + b 2 ( x ) x 2 z 2 ( 1 − x z ) ( 1 − x 2 z ) ( 1 − x 3 z ) ⋯ {\displaystyle {\begin{aligned}S(x,z)=&{\frac {S(x,zx)}{1-xz}}\\=&{\frac {1}{1-xz}}\left(1+{\frac {b_{1}(x)xz}{1-x^{2}z}}+{\frac {b_{2}(x)x^{2}z^{2}}{(1-x^{2}z)(1-x^{3}z)}}+\cdots \right)\\=&{\frac {1}{1-xz}}+{\frac {b_{1}(x)xz}{(1-xz)(1-x^{2}z)}}+{\frac {b_{2}(x)x^{2}z^{2}}{(1-xz)(1-x^{2}z)(1-x^{3}z)}}\cdots \end{aligned}}}
であるが