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Quintuple product identity

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(Redirected from Watson's quintuple product identity) Infinite product identity introduced by Watson

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

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.

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