The Jacobi triple product is the beautiful identity product_(n = 1)^∞(1 - x^(2n))(1 + x^(2n - 1) z^2)(1 + x^(2n - 1)/z^2) = sum_(m = - ∞)^∞ x^(m^2) z^(2m). In terms of the Q-functions, (-1) is written Q_1 Q_2 Q_3 = 1, which is one of the two Jacobi identities. In q-series notation, the Jacobi triple product identity is written (q, - x q, -1/x;q)_∞ = sum_(k = - ∞)^∞ x^k q^(k(k + 1)/2) for 0< left bracketing bar q right bracketing bar <1 and x!=0. Another form of the identity is sum_(n = - ∞)^∞ (-1)^n a^n q^(n(n - 1)/2) = product_(n = 1)^∞(1 - a q^(n - 1))(1 - a^(-1) q^n)(1 - q^n) (Hirschhorn 1999).
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