In mathematics the Watson quintuple product identity is an infinite product identity introduced by Watson (1929) and rediscovered by Bailey (1951) and 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
∏ n ≥ 1 ( 1 − s n ) ( 1 − s n t ) ( 1 − s n − 1 t − 1 ) ( 1 − s 2 n − 1 t 2 ) ( 1 − s 2 n − 1 t − 2 ) = ∑ n ∈ Z s ( 3 n 2 + n ) / 2 ( t 3 n − t − 3 n − 1 ) {\displaystyle \prod _{n\geq 1}(1-s^{n})(1-s^{n}t)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^{2})(1-s^{2n-1}t^{-2})=\sum _{n\in \mathbf {Z} }s^{(3n^{2}+n)/2}(t^{3n}-t^{-3n-1})}
References 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 Carlitz, L.; Subbarao, M. V. (1972), "A simple proof of the quintuple product identity", Proceedings of the American Mathematical Society, 32 (1): 42–44, doi:10.2307/2038301, ISSN 0002-9939, JSTOR 2038301, MR 0289316 Gordon, Basil (1961), "Some identities in combinatorial analysis", The Quarterly Journal of Mathematics, Second Series, 12: 285–290, doi:10.1093/qmath/12.1.285, ISSN 0033-5606, MR 0136551 Watson, G. N. (1929), "Theorems stated by Ramanujan. VII: Theorems on continued fractions.", Journal of the London Mathematical Society, 4 (1): 39–48, doi:10.1112/jlms/s1-4.1.39, ISSN 0024-6107, JFM 55.0273.01 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 Hirschhorn–Farkas–Kra septagonal numbers identity
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.