Product of power series

Builds on Formal power series Coefficient of a power series

Statement

Let $R$ be a commutative ring, let $f, g \in R[\![X]\!]$ be formal power series, and let $n$ be a natural number. Then $[X^n](f g) = \sum_{i + j = n} [X^i] f \cdot [X^j] g = \sum_{i=0}^{n} [X^i] f \cdot [X^{n-i}] g.$

Remarks

In many texts this formula is the definition of the product of formal power series. The sequence of coefficients of $fg$ is the convolution of the sequences of $f$ and $g$ . This is the Cauchy product. Example: $[X^n] \frac{1}{(1 - X)^2} = \sum_{i=0}^{n} 1 \cdot 1 = n + 1$ .

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Formal power series definition
Coefficient of a power series definition
Product of power series statement

Tag	`0H6KWAYS87`
URI	`m2://0H6KWAYS87`
Kind	statement, theorem
Writing	W0 stub
MSC	13F25
Status	This copy of the site has no build data. Run `m2 status 0H6KWAYS87`.
Uses	Formal power series, Coefficient of a power series

Product of power series

Statement

Remarks

Reading path

Further reading

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