「ワトソンの五重積」の版間の差分
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m Category:エポニムを除去; Category:数学のエポニムを追加 (HotCat使用) |
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==参考文献== |
==参考文献== |
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* 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 |
* 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 |
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* Gordon, Basil (1961), "Some identities in combinatorial analysis", The Quarterly Journal of Mathematics. Oxford. Second Series, 12: 285–290, doi |
* 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 |
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* Carlitz, L.; Subbarao, M. V. (1972), "A simple proof of the quintuple product identity", Proceedings of the American Mathematical Society, 32: 42–44, doi |
* 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 |
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* Foata, |
* {{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}} |
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* Cooper, |
* {{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}} |
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==関連文献== |
==関連文献== |
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===和文=== |
===和文=== |
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* {{PDFlink|[ |
* {{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) |
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* On Jacobi forms which connect infinite products and infinite sums, |
* {{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}} |
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* {{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) |
* {{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) |
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* Hirschhorn, M. D. (1988). A generalisation of the quintuple product identity. Journal of the Australian Mathematical Society, 44(1), 42-45. |
* Hirschhorn, M. D. (1988). A generalisation of the quintuple product identity. Journal of the Australian Mathematical Society, 44(1), 42-45. |
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* Alladi, K. (1996). The quintuple product identity and shifted partition functions. [[:en:Journal of Computational and Applied Mathematics]], 68(1-2), 3-13. |
* Alladi, K. (1996). The quintuple product identity and shifted partition functions. [[:en:Journal of Computational and Applied Mathematics]], 68(1-2), 3-13. |
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* Farkas, |
* {{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/}} |
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* Chen, |
* {{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}} |
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* Chu, W., & Yan, Q. (2007). Unification of the quintuple and septuple product identities. The electronic journal of combinatorics. |
* Chu, W., & Yan, Q. (2007). Unification of the quintuple and septuple product identities. The electronic journal of combinatorics. |
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{{DEFAULTSORT:わとそんのこしゆうせき}} |
{{DEFAULTSORT:わとそんのこしゆうせき}} |
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2024年2月6日 (火) 06:25時点における最新版
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。
証明[編集]
ヤコビの三重積により
オイラーの五角数定理により
これらを用いて五重積の公式を書き直せば
となるので、この両辺が等しいことを証明する。左辺は
さて
であるから
となり、右辺を得る。
関連項目[編集]
- ジョージ・ネビル・ワトソン (en)
- ヤコビの三重積
- ファルカシュの七重積 (英: septuple product identity)[1][2]
出典[編集]
参考文献[編集]
- 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
- Foata, Dominique; Han, Guo-Niu (2001). The triple, quintuple and septuple product identities revisited. The Andrews festschrift: seventeen papers on classical issue theory and combinatorics. pp. 323-334. doi:10.1007/978-3-642-56513-7_15. ISBN 978-3-642-56513-7 .
- Cooper, Shaun (2006). “The quintuple product identity”. International Journal of Number Theory (World Scientific) 2 (01): 115-161. doi:10.1142/S1793042106000401 .
関連文献[編集]
和文[編集]
- 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)
- 青木宏樹「On Jacobi forms which connect infinite products and infinite sums (Research on finite groups and their representations, vertex operator algebras, and algebraic combinatorics)」『数理解析研究所講究録』第1965巻、京都大学数理解析研究所、2015年10月、30-44頁、CRID 1050282810823192320、hdl:2433/224233、ISSN 1880-2818“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)
英文[編集]
- 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.
- Farkas, Hershel; Kra, Irwin (1999). “On the quintuple product identity”. Proceedings of the American Mathematical Society 127 (3): 771-778 .
- Chen, William YC; Chu, Wenchang; Gu, Nancy SS (2005). “Finite form of the quintuple product identity”. arXiv preprint 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.