How to evaluate this troublesome limit?

Question

I recently encountered the following limit :

$$\displaystyle\lim_{p\to\infty}\dfrac{c^{(p^2+p)/2}}{\displaystyle\prod_{k=1}^p‌(c^k-1)},c>1$$

How can one evaluate it ? (from the comments it can be shown that it converges to a real value, but what is its limit ?)

It's the same as asking for the limit of $$\prod_{k=1}^p \frac{1}{1-c^{-k}}$$ — Thomas Andrews, Jan 21 '15 at 13:31

score 2 · Accepted Answer · answered Jan 21 '15 at 13:37

Writing $c^k-1=c^k(1-c^{-k})$, the formula is:

$$\prod_{k=1}^\infty\frac{1}{1-c^{-k}}$$

The function, the reciprocal of Euler's function:

$$f(x)=\prod_{k=1}^\infty\frac{1}{1-x^k}$$

is the generating function for the partition function, and the limit you are asking for is $f\left(\frac{1}{c}\right)$.

There is no known closed formula.

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