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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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    Btw, I found that in Mathematica it can be represented as $\operatorname{QPochhammer}[q^n]$2011-02-01

2 Answers 2

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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 $\sin(x)$.