ワトソンの五重積のソースを表示

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。
:<math>\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})</math>

== 証明 ==
[[ヤコビの三重積]]により
:<math>\sum_{n=-\infty}^{\infty}(-1)^nq^{n(n+1)}z^{n}=\prod_{m=1}^{\infty}(1-q^{2m})(1-q^{2m}z)(1-q^{2m-2}z^{-1})</math>
:<math>\sum_{n=-\infty}^{\infty}(-1)^nq^{2n^2}z^{2n}=\prod_{m=1}^{\infty}(1-q^{4m})(1-q^{4m-2}z^{2})(1-q^{4m-2}z^{-2})</math>
[[オイラーの五角数定理]]により
:<math>\sum_{n=-\infty}^{\infty}(-1)^nq^{2n(3n-1)}=\prod_{m=1}^{\infty}(1-q^{4m})</math>

これらを用いて五重積の公式を書き直せば
:<math>\sum_{m=-\infty}^{\infty}(-1)^mq^{2m(3m-1)}\sum_{n=-\infty}^{\infty}q^{n(3n+1)}(z^{3n}-z^{-3n-1})=\sum_{j=-\infty}^{\infty}(-1)^jq^{j(j+1)}z^{j}\sum_{k=-\infty}^{\infty}(-1)^kq^{2k^2}z^{2k}</math>
となるので、この両辺が等しいことを証明する。左辺は
:<math>\begin{align}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{align}</math>
さて
:<math>\begin{align}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{align}</math>
であるから
:<math>\begin{align}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{align}</math>
となり、右辺を得る。

== 関連項目 ==
* [[ジョージ・ネビル・ワトソン]] ([[:en:G.N. Watson|en]])
* [[ヤコビの三重積]]
* [[ファルカシュの七重積]] ({{lang-en-short|septuple product identity}})<ref>Yan, Q. (2009). A new proof of the septuple product identity. Discrete Mathematics, 309(8), 2589-2591.</ref><ref>Chapman, R. (1999). On a septuple product identity.</ref>

==出典==
{{reflist}}

==参考文献==
* Bailey, W. N. (1951), "On the simplification of some identities of the Rogers-Ramanujan type", Proceedings of the London Mathematical Society, Third Series, 1: 217–221, {{doi|10.1112/plms/s3-1.1.217}}, {{ISSN|0024-6115}}, MR 0043839
* Gordon, Basil (1961), "Some identities in combinatorial analysis", The Quarterly Journal of Mathematics. Oxford. Second Series, 12: 285–290, {{doi|10.1093/qmath/12.1.285}}, {{ISSN|0033-5606}}, MR 0136551
* Carlitz, L.; Subbarao, M. V. (1972), "A simple proof of the quintuple product identity", Proceedings of the American Mathematical Society, 32: 42–44, {{doi|10.2307/2038301}}, {{ISSN|0002-9939}}, {{JSTOR|2038301}}, MR 0289316
* {{Cite journal |author=Foata, Dominique; Han, Guo-Niu |year=2001 |url=https://doi.org/10.1007/978-3-642-56513-7_15 |title=The triple, quintuple and septuple product identities revisited |series=The Andrews festschrift: seventeen papers on classical issue theory and combinatorics |pages=323-334 |organization=Springer |doi=10.1007/978-3-642-56513-7_15 |ISBN=978-3-642-56513-7}}
* {{Cite journal |author=Cooper, Shaun |year=2006 |title=The quintuple product identity |journal=International Journal of Number Theory |volume=2 |issue=01 |pages=115-161 |url=https://doi.org/10.1142/S1793042106000401 |publisher=World Scientific |doi=10.1142/S1793042106000401}}

==関連文献==
===和文===
* {{PDFlink|[https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kokai-koza/H29-toshiya.pdf 五重積公式のADE一般化—場の理論の視点から—河合俊哉]}}, 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)
* {{Cite journal|和書|author=青木宏樹 |year=2015 |month=10 |url=https://hdl.handle.net/2433/224233 |title=On Jacobi forms which connect infinite products and infinite sums (Research on finite groups and their representations, vertex operator algebras, and algebraic combinatorics) |journal=数理解析研究所講究録 |ISSN=1880-2818 |publisher=京都大学数理解析研究所 |volume=1965 |pages=30-44 |hdl=2433/224233 |CRID=1050282810823192320 |quote=五重積公式について解説がある}}
* {{PDFlink|[https://www.ma.noda.tus.ac.jp/u/ha/Data/kyushu.pdf A remark on Borcherds construction of Jacobi forms]}} (五重積公式について解説がある)

===英文===
* Subbarao, M. V., & Vidyasagar, M. (1970). On Watson’s quintuple product identity. Proceedings of the American Mathematical Society, 26(1), 23-27.
* Hirschhorn, M. D. (1988). A generalisation of the quintuple product identity. Journal of the Australian Mathematical Society, 44(1), 42-45.
* Alladi, K. (1996). The quintuple product identity and shifted partition functions. [[:en:Journal of Computational and Applied Mathematics]], 68(1-2), 3-13.
* {{Cite journal |author=Farkas, Hershel; Kra, Irwin |year=1999 |title=On the quintuple product identity |journal=Proceedings of the American Mathematical Society |volume=127 |issue=3 |pages=771-778 |url=https://www.ams.org/journals/proc/1999-127-03/S0002-9939-99-04791-7/}}
* {{Cite journal |author=Chen, William YC; Chu, Wenchang; Gu, Nancy SS |year=2005 |title=Finite form of the quintuple product identity |journal=arXiv preprint math/0504277 |url=https://doi.org/10.48550/arXiv.math/0504277 |doi=10.48550/arXiv.math/0504277}}
* Chu, W., & Yan, Q. (2007). Unification of the quintuple and septuple product identities. The electronic journal of combinatorics.

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[[Category:楕円函数論]]
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[[Category:恒等式]]
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