In mathematics the Watson quintuple product identity is an infinite product identity introduced by and rediscovered by and . It is analogous to the

Jacobi triple product identity In mathematics, the Jacobi triple product is the mathematical identity: :\prod_^\infty \left( 1 - x^\right) \left( 1 + x^ y^2\right) \left( 1 +\frac\right) = \sum_^\infty x^ y^, for complex numbers ''x'' and ''y'', with , ''x'', < 1 and ''y ...

, and is the Macdonald identity for a certain non-reduced

affine root system In mathematics, an affine root system is a root system of affine-linear functions on a Euclidean space. They are used in the classification of affine Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g to ...

. It is related to Euler's

pentagonal number theorem In mathematics, the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function. It states that :\prod_^\left(1-x^\right)=\sum_^\left(-1\right)^x^=1+\sum_^\infty(-1)^k\left(x^+x^\righ ...

Statement

\prod_ (1-s^n)(1-s^nt)(1-s^t^)(1-s^t^2)(1-s^t^) 
= \sum_s^(t^-t^)

References

* * * *{{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.

* 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. Elliptic functions Theta functions Mathematical identities Theorems in number theory Infinite products

Statement

References

Further reading