ワトソンの五重積

数学において次の恒等式をワトソンの五重積 (ワトソンのごじゅうせき、Watson Quintuple Product) という。

\sum _{n=-\infty }^{\infty }q^{n(3n+1)}(z^{3n}-z^{-3n-1})=\prod _{m=1}^{\infty }(1-q^{2m})(1-q^{2m}z)(1-q^{2m-2}z^{-1})(1-q^{4m-2}z^{2})(1-q^{4m-2}z^{-2})

証明

\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n(n+1)}z^{n}=\prod _{m=1}^{\infty }(1-q^{2m})(1-q^{2m}z)(1-q^{2m-2}z^{-1})

\sum _{n=-\infty }^{\infty }(-1)^{n}q^{2n^{2}}z^{2n}=\prod _{m=1}^{\infty }(1-q^{4m})(1-q^{4m-2}z^{2})(1-q^{4m-2}z^{-2})

\sum _{n=-\infty }^{\infty }(-1)^{n}q^{2n(3n-1)}=\prod _{m=1}^{\infty }(1-q^{4m})

これらを用いて五重積の公式を書き直せば

\sum _{m=-\infty }^{\infty }(-1)^{m}q^{2m(3m-1)}\sum _{n=-\infty }^{\infty }q^{n(3n+1)}(z^{3n}-z^{-3n-1})=\sum _{j=-\infty }^{\infty }(-1)^{j}q^{j(j+1)}z^{j}\sum _{k=-\infty }^{\infty }(-1)^{k}q^{2k^{2}}z^{2k}

となるので、この両辺が等しいことを証明する。左辺は

{\begin{aligned}L&=\sum _{m=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }(-1)^{m}q^{2m(3m-1)+n(3n+1)}(z^{3n}-z^{-3n-1})\\&=\sum _{m=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }(-1)^{m}q^{2m(3m-1)+n(3n+1)}z^{3n}-\sum _{m=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }(-1)^{m}q^{-2m(-3m-1)-n(-3n+1)}z^{3n-1}\qquad ({m,n}\mapsto {-m,-n})\\&=\sum _{k=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }(-1)^{k-n}q^{(3n-2k)(3n+1-2k)+2k^{2}}z^{3n}-\sum _{k=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }(-1)^{k-n}q^{(3n-1-2k)(3n-2k)+2k^{2}}z^{3n-1}\qquad (m\mapsto {k-n})\\&=\sum _{k=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }(-1)^{3n-k}q^{(3n-2k)(3n+1-2k)+2k^{2}}z^{3n}+\sum _{k=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }(-1)^{3n-1-k}q^{(3n-1-2k)(3n-2k)+2k^{2}}z^{3n-1}\\\end{aligned}}

さて

{\begin{aligned}0&=\sum _{m=0}^{\infty }(-1)^{m}q^{m(m+1)}-\sum _{n=0}^{\infty }(-1)^{n}q^{n(n+1)}=\sum _{m=-\infty }^{\infty }(-1)^{m}q^{m(m+1)}\qquad (n\mapsto {-(m+1)})\\&=\sum _{m=-\infty }^{\infty }(-1)^{m}(q^{6})^{m(m-1)}\sum _{n=-\infty }^{\infty }(-1)^{2n-2}q^{3n^{2}-3n+2}z^{3n-2}\\&=\sum _{m=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }(-1)^{m+2n-2}q^{6m(m-1)+3n^{2}-3n+2}z^{3n-2}\\&=\sum _{k=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }(-1)^{3n-2-k}q^{(3n-2-2k)(3n-1-2k)+2j^{2}}z^{3n-2}\qquad (m\mapsto {n-k})\\\end{aligned}}

であるから

{\begin{aligned}L=L+0&=\sum _{k=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }(-1)^{n-k}q^{(n-2k)(n+1-2k)+2k^{2}}z^{n}\qquad ({3n,3n-1,3n-2}\mapsto {n})\\&=\sum _{k=-\infty }^{\infty }\sum _{j=-\infty }^{\infty }(-1)^{k+j}q^{j(j+1)+2k^{2}}z^{j+2k}\qquad ({n}\mapsto {j+2k})\\&=\sum _{j=-\infty }^{\infty }(-1)^{j}q^{j(j+1)}z^{j}\sum _{k=-\infty }^{\infty }(-1)^{k}q^{2k^{2}}z^{2k}\\\end{aligned}}

となり、右辺を得る。

^ Yan, Q. (2009). A new proof of the septuple product identity. Discrete Mathematics, 309(8), 2589-2591.
^ Chapman, R. (1999). On a septuple product identity.

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五重積公式のADE一般化—場の理論の視点から—河合俊哉 (PDF) , 平成29年度 (第39回) 数学入門公開講座テキスト (京都大学数理解析研究所，平成29年7月31日～8月3日開催)
On Jacobi forms which connect infinite products and infinite sums, 青木宏樹, 京都大学数理解析研究所講究録第1965巻 2015年 30-44. (五重積公式について解説がある)
A remark on Borcherds construction of Jacobi forms (PDF) (五重積公式について解説がある)

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