Expansion in power series of $z=0$

Asked Apr 26 '17 at 14:38

Active Apr 26 '17 at 14:46

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How do I expand this function in power series?

$$f(x):= \frac{1}{1+x+x^2+x^3}.$$

I have tried to factorize it and we have

$$f(x)= \frac{1}{2}\left(\frac{1}{x+1}-\frac{-1+x}{x^2+1}\right)$$

however I am not sure if it leads anywhere, because the other expression has an $x$ in the counter and I do not know how to use $1/(1-x)$ series expansion.

edited Apr 26 '17 at 14:46

asked Apr 26 '17 at 14:38

Ryszard Eggink

1

Hint: $f(x)=\frac{1-x}{1-x^4}=(1-x)\sum_{k\geq 0} x^{4k}$ – Kelenner Apr 26 '17 at 14:46
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$\frac{1}{1+x}$ is the generating function of $1,-1,1,-1,1,-1,\ldots$ while $\frac{1}{1+x^2}$ is the generating function of $1,0,-1,0,1,0\ldots$ and $\frac{x}{1+x^2}$ is the generating function of $0,1,0,-1,0,1,\ldots$. So add them together and divide by $2$ – Henry Apr 26 '17 at 14:46
Kelenner, is that a well known fact? Could you explain where it comes from? – Ryszard Eggink Apr 26 '17 at 14:59
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@RyszardEggink Yes, it is a well known fact – A.Γ. Apr 26 '17 at 15:02

Expansion in power series of $z=0$

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