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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?

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2  
en.wikipedia.org/wiki/Euler_function some sparse info –  InterestedGuest Feb 1 '11 at 19:25
    
Btw, I found that in Mathematica it can be represented as $\operatorname{QPochhammer}[q^n]$ –  Anixx Feb 1 '11 at 19:33

2 Answers 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.

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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 $\sin(x)$.

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