Math 2.0

statement theorem

Product of power series

Statement

Let RR be a commutative ring, let f,g∈R[ ⁣[X] ⁣]f, g \in R[\![X]\!] be formal power series, and let nn be a natural number. Then [Xn](fg)=∑i+j=n[Xi]f⋅[Xj]g=∑i=0n[Xi]f⋅[Xn−i]g.[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 fgfg is the convolution of the sequences of ff and gg. This is the Cauchy product. Example: [Xn]1(1−X)2=∑i=0n1⋅1=n+1[X^n] \frac{1}{(1 - X)^2} = \sum_{i=0}^{n} 1 \cdot 1 = n + 1.

Explanations

L1 human

To find the coefficient of XnX^n in a product, collect all ways to get XnX^n as XiX^i times XjX^j. Add the products of the matching coefficients.

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Formal power series, definition, no evidence levelFormal power seriesdefinitionCoefficient of a power series, definition, no evidence levelCoefficient of a powe…definitionProduct of power series, statement, no evidence levelProduct of power seri…statement
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  1. Formal power series definition
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Sources

  1. Web https://en.wikipedia.org/wiki/Cauchy_product, Cauchy product of two power series

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