In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 In mathematics, the Macdonald identities are some infinite product identities associated to affine root systems, introduced by . They include as special cases the Jacobi triple product identity, Watson's quintuple product identity, several identiti ...
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 algebras and superalgebras, and semisimple ''p''-adic algebraic groups, and correspond to f ...
. 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^\right ...
.
Statement
:
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.
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
The ''Journal of Computational and Applied Mathematics'' is a peer-reviewed scientific journal covering computational and applied mathematics. It was established in 1975 and is published biweekly by Elsevier. The editors-in-chief are Yalchin Efendi ...
, 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