4
$\begingroup$

I'd like to know what $\prod_{k=1}^n (1-x^k)$

evaluates to (assuming there is a simple closed form) and what it "is" in the context of commutative algebra (of which I knew little and recall less).

I'm sure I've seen this in the past but don't know where to place it. LaTeX search doesn't help.

  • 1
    Also, there's a sum on the mathworld page for the finite q-pochhammer symbol that gives it in terms of the q-binomial (equation 4)2011-03-30

1 Answers 1

7

Well, one has $\prod_{n\geq1}(1-x^k) = \sum_{-\infty\leq n\leq\infty}(-1)^nx^{(3n^2-n)/2}.$ This is a consequence of Jacobi's triple product identity.

You asked about a finite product, but from this equality you can tell what are the coefficients in the expanded finite product.

The context for this identities is, among others, the theory of partitions. I am sure you will find a proof of this in Andrews' excellent The Theory of Partitions, along with related information.

  • 0
    @SHuntsman I used the mentioned reciprocal to answer one of my questions on "Number of partitions of 2n with no element greater than n" [here](http://math.stackexchange.com/a/107169/19341). Did you have a comparable problem?2012-08-03