How can one compute this simple infinite product?

Question

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Limit of a particular variety of infinite product/series

Define $$F(x) = \prod_{n=1}^\infty(1-x^n)$$ where $|x|<1$.

How can one compute $F(1/2)$? (Without an obvious polynomial expansion or brute-force calculation.)

This is sometimes called Euler's function.

@RaymondManzoni that's a great source. Nearly as good as a closed form. — Pricklebush Tickletush, Dec 13 '12 at 18:10
Glad it helped @AlecS but I fear we will have to close this as duplicate! — Raymond Manzoni, Dec 13 '12 at 18:16

score 3 · Accepted Answer · edited Dec 13 '12 at 18:44

3

That product is not so simple as you think. Euler proved that $$F(x) = \prod_{n=1}^\infty(1-x^n)=\sum_{-\infty}^{+\infty}(-1)^kx^{\frac{k(3k+1)}{2}}$$ This problem arises in partition number theory and is called Euler's pentagonal number theorem.

edited Dec 13 '12 at 18:44

Pricklebush Tickletush

344

answered Dec 13 '12 at 18:17

Adi Dani

16,949

@AlecS: This paper from Bell may help you too. – Raymond Manzoni Dec 13 '12 at 18:29
@RaymondManzoni I had begun reading it already. :) – Pricklebush Tickletush Dec 13 '12 at 18:36
@AlecS: a rather fascinating subject I'll admit... – Raymond Manzoni Dec 13 '12 at 18:45

How can one compute this simple infinite product?

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