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Given two power series,

$$f(x)=\sum_{n=0}^{\infty}a_{n}x^{n}$$

and

$$g(x)=\sum_{n=0}^{\infty}b_{n}x^{n}.$$

It is easy to form their product

$$f(x)g(x)=\sum_{n=0}^{\infty}c_{n}x^{n}$$

where

$$c_{n}=\sum_{k=0}^{n}a_{k}b_{n-k}.$$

But many of the series I come across only contain negative powers of $x$, that is

$$h(x)=\sum_{n=0}^{\infty}d_{n}x^{-n}.$$

Is there any tricks or methods anyone knows of to find the series representation of the product $f(x)h(x)$ ??

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    Formally, I think this is just a matter of changing $x$ into $\frac{1}{x}$ in your formula.2011-06-11
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    @Joel Cohen But if it was a simple matter of making the substitution $x\mapsto\frac{1}{x}$ the resulting power series would be for the product $f(x)h(\frac{1}{x})$. I dont see an obvious way to transform the this series to obtain a series for $f(x)h(x)$2011-06-12

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