Euler's pentagonal number theorem

product_(k = 1)^∞(1 - x^k) | = | sum_(k = - ∞)^∞ (-1)^k x^(k(3k + 1)/2) | = | 1 + sum_(k = 1)^∞ (-1)^k[x^(k(3k - 1)/2) + x^(k(3k + 1)/2)] | = | (x)_∞ | = | 1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 + x^26 - ... (OEIS A010815), where 0, 1, 2, 5, 7, 12, 15, 22, 26, ... (OEIS A001318) are generalized pentagonal numbers and (x)_∞ is a q-Pochhammer symbol. This identity was proved by Euler in a paper presented to the St. Petersburg Academy on August 14, 1775.

partition function P | partition function Q | pentagonal number | q-Pochhammer symbol | Ramanujan theta functions | Zagier's identity