Quintuple product identity explained

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, and is the Macdonald identity for a certain non-reduced affine root system. It is related to Euler's pentagonal number theorem.

Statement

\prod _{n\ge} ( 1- s ^{n)(1-s} ^{nt)(1-s} ^{n-1} t ^{-1} )( 1- s ^{2n-1} t ^{2)(1-s} ^{2n-1} t ^{-2} ) = \sum _{n\in} s ^{( 3 n ^{2+n)/2}} (t ^{3n} - t ^{-3n-1} )

References

• • • • • 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, 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.