What is Eulerian?

Question

I encountered an interesting function which is called "Eulerian" by the Wolfram's MathWorld:

$$\phi(q)=\prod_{k=1}^{\infty} (1-q^{k})$$

It is interesting because it seems that roots of any polynomial can be expressed in this function and elementary functions.

I want to know more about the properties of this function, where can I find the information?

Answer 1

You may want to look up Weierstrass factorization theorem which plays a crucial role in complex analysis for writing functions as infinite products. It is a simple but powerful idea. Euler's famous proof of the Basel's problem exploited this infinite product for $\sin(x)$.

Answer 2

One notable property of this function is Euler's Pentagonal Number Theorem:

$$\prod_{k=1}^\infty (1-x^k) = \sum_{k=-\infty}^\infty (-1)^k x^{k(3k-1)/2}.$$

Here is a very interesting paper on the Pentagonal Number Theorem by Jordan Bell.