Pentagonal number theorem

In mathematics, the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function. It states that

In other words,

The exponents 1, 2, 5, 7, 12, ... on the right hand side are given by the formula g_k = k(3k-1)/2 for k = 1, −1, 2, −2, 3, ... and are called (generalized) pentagonal numbers. This holds as an identity of convergent power series for ${\displaystyle |x|<1}$, and also as an identity of formal power series.

A striking feature of this formula is the amount of cancellation in the expansion of the product.

The identity implies a marvelous recurrence for calculating ${\displaystyle p(n)}$, the number of partitions of n:

or more formally,

where the summation is over all nonzero integers k (positive and negative) and ${\displaystyle g_{k}}$ is the k^th generalized pentagonal number.

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