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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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    @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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If it exists, then we can say that the function is defined using the infinite Laurent series $f(z)h(z) = \sum_{j=-\infty}^{\infty} c_j z^j$ where $c_j$ is defined for nonnegative $j$ as $c_j=\sum_{k=0}^{\infty} a_k d_{k+j}$ and for nonpositive $j$ as

$c_j=\sum_{k=0}^{\infty} a_{k-j} d_k$

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    @Robert Israel. I want to find the series representation of the product $fh$. The series representation of $h$ consists only of terms involving non-positive integer powers of $x$ and $f$ has a power series representation. If I treat them both as Laurent series I have little hope of determining the coefficients $c_n$. Thats the point i was trying to make. Perhaps i'm asking for something that does not exist but i was hoping i could compute the coefficients $c_n$ from $a_n$ and $d_n$. As opposed to dealing with infinite sums2011-06-13