Watson quintuple product identity

The quintuple product identity, also called the Watson quintuple product identity, states product_(n = 1)^∞(1 - q^n)(1 - z q^n)(1 - z^(-1) q^(n - 1))(1 - z^2 q^(2n - 1))(1 - z^(-2) q^(2n - 1)) = sum_(m = - ∞)^∞(z^(3m) - z^(-3 m - 1)) q^(m(3m + 1)/2). It can also be written product_(n = 1)^∞(1 - q^(2n))(1 - q^(2n - 1) z)(1 - q^(2n - 1) z^(-1))(1 - q^(4n - 4) z^2)(1 - q^(4n - 4) z^(-2)) = sum_(n = - ∞)^∞ q^(3n^2 - 2n)[(z^(3n) + z^(-3 n)) - (z^(3n - 2) + z^(-(3n - 2)))] or sum_(k = - ∞)^∞ (-1)^k q^((3k^2 - k)/2) z^(3k)(1 + z q^k) = product_(j = 1)^∞(1 - q^j)(1 + z^(-1) q^j)(1 + z q^(j - 1))(1 + z^(-2) q^(2j - 1))(1 + z^2 q^(2j - 1)).

Jacobi triple product | Ramanujan theta functions