Pentagonal number theorem
Theorem in number theory / From Wikipedia, the free encyclopedia
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In mathematics, Euler's pentagonal number theorem 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 gk = k(3k − 1)/2 for k = 1, −1, 2, −2, 3, ... and are called (generalized) pentagonal numbers (sequence A001318 in the OEIS). (The constant term 1 corresponds to .) This holds as an identity of convergent power series for , 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.