Quintuple product identity

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} (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 \mathbf{Z}}s^{(3n^2+n)/2}(t^{3n}-t^{-3n-1}) $$