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I have $n$ power series. How can I find the power series of the product of these $n$ series? If there are two series $(a_m)$ and $(b_m)$ then the product series $(c_m)$ is given by the Cauchy product,

$c_m = \sum_{k=0}^m a_k b_{m-k}$

How does this generalize to more series?

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    @Pete @Chris, cool then my bad. I should have commented that I could'nt make sense of it and nothing to$w$ards the OP.2011-06-15

2 Answers 2

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$\left(\sum_{n=0}^\infty a_{1,n} x^n\right)\left(\sum_{n=0}^\infty a_{2,n} x^n\right)\dots\left(\sum_{n=0}^\infty a_{l,n} x^n\right) = \sum_{n=0}^\infty \sum_{k_1 + k_2 + \dots + k_l = n } a_{1,k_1}a_{2,k_2}\dots a_{l,k_l} x^n $

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Presumably you are looking for an expression for the coefficients of $x^i$ in the product

$\left(\sum_j a_j x^j \right) \left(\sum_k b_k x^k \right) \left(\sum_l c_l x^l \right) \ldots \left(\sum_m n_m x^m \right)$

Generalising the Cauchy product, this is

$\sum_i \left(\sum_{j=0}^{i} \sum_{k=0}^{i-j} \sum_{l=0}^{i-j-k} \cdots a_j b_k c_l \ldots n_{i-j-k-l-\cdots} \right) x^i$