ヤコビの三重積
Θ ( v + τ , τ ) = ∏ 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 − 3 ) π i τ − 2 π i v ) = 1 + e − π i τ − 2 π i v 1 + e π i τ + 2 π i 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 ) = e − π i τ − 2 π i v + 1 1 + e π i τ + 2 π i 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 ) = e − π i τ − 2 π i v Θ ( v , τ ) {\displaystyle {\begin{aligned}\Theta (v+\tau ,\tau )&=\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-3){\pi }i{\tau }-2{\pi }iv}\right)}\\&={\frac {1+e^{-{\pi }i{\tau }-2{\pi }iv}}{1+e^{{\pi }i{\tau }+2{\pi }iv}}}\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)}\\&={\frac {e^{-{\pi }i{\tau }-2{\pi }iv}+1}{1+e^{{\pi }i{\tau }+2{\pi }iv}}}\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)}\\&=e^{-{\pi }i{\tau }-2{\pi }iv}\Theta (v,\tau )\end{aligned}}}
さて、ヤコビの三重積の冪級数展開を得たいが、代わりに R ( x , z ) R ( x , z − 1 ) {\displaystyle R(x,z)R(x,z^{-1})} の冪級数展開について考える。
R ( x , z ) R ( x , z − 1 ) = ( 1 + a 1 ( x ) z + a 2 ( x ) z 2 + ⋯ ) ( 1 + a 1 ( x ) z − 1 + a 2 ( x ) z − 2 + ⋯ ) {\displaystyle R(x,z)R(x,z^{-1})=(1+a_{1}(x)z+a_{2}(x)z^{2}+\cdots )(1+a_{1}(x)z^{-1}+a_{2}(x)z^{-2}+\cdots )}
より R ( x , z ) R ( x , z − 1 ) {\displaystyle R(x,z)R(x,z^{-1})} を冪級数展開したときの z m {\displaystyle z^{m}} および z − m {\displaystyle z^{-m}} の係数は共に