Quintuple product identity
{{short description|Infinite product identity introduced by Watson}}
In mathematics the Watson quintuple product identity is an infinite product identity introduced by {{harvs|txt|authorlink=G. N. Watson|last=Watson|year=1929}} and rediscovered by {{harvtxt|Bailey|1951}} and {{harvtxt|Gordon|1961}}. It is analogous to the Jacobi triple product identity, and is the Macdonald identity for a certain non-reduced affine root system. It is related to Euler's pentagonal number theorem.
Statement
:
= \sum_{n\in \mathbf{Z}}s^{(3n^2+n)/2}(t^{3n}-t^{-3n-1})
References
- {{Citation | last1=Bailey | first1=W. N. | title=On the simplification of some identities of the Rogers-Ramanujan type | doi=10.1112/plms/s3-1.1.217 | mr=0043839 | year=1951 | journal=Proceedings of the London Mathematical Society |series=Third Series | issn=0024-6115 | volume=1 | pages=217–221}}
- {{Citation | last1=Carlitz | first1=L. | author1-link=Leonard Carlitz | last2=Subbarao | first2=M. V. | title=A simple proof of the quintuple product identity | jstor=2038301 | mr=0289316 | year=1972 | journal=Proceedings of the American Mathematical Society | issn=0002-9939 | volume=32 | issue=1 | pages=42–44 | doi=10.2307/2038301| doi-access=free }}
- {{Citation | last1=Gordon | first1=Basil | author1-link=Basil Gordon | title=Some identities in combinatorial analysis | mr=0136551 | year=1961 | journal=The Quarterly Journal of Mathematics |series=Second Series | issn=0033-5606 | volume=12 | pages=285–290 | doi=10.1093/qmath/12.1.285}}
- {{Citation | last1=Watson | first1=G. N. | title=Theorems stated by Ramanujan. VII: Theorems on continued fractions. | doi=10.1112/jlms/s1-4.1.39 | jfm=55.0273.01 | year=1929 | journal=Journal of the London Mathematical Society | issn=0024-6107 | volume=4 | issue=1 | pages=39–48}}
- Foata, D., & Han, G. N. (2001). The triple, quintuple and septuple product identities revisited. In The Andrews Festschrift (pp. 323–334). Springer, Berlin, Heidelberg.
- Cooper, S. (2006). The quintuple product identity. International Journal of Number Theory, 2(01), 115-161.
See also
Further reading
- 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. Journal of Computational and Applied Mathematics, 68(1-2), 3-13.
- Farkas, H., & Kra, I. (1999). On the quintuple product identity. Proceedings of the American Mathematical Society, 127(3), 771-778.
- Chen, W. Y., Chu, W., & Gu, N. S. (2005). Finite form of the quintuple product identity. arXiv preprint math/0504277.
Category:Mathematical identities