Binomial Coefficients and Generating Functions - Cs.ioc.ee
Binomial Coefficients and Generating Functions - Cs.ioc.ee
Binomial Coefficients and Generating Functions - Cs.ioc.ee
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Operations on <strong>Generating</strong> <strong>Functions</strong> (6)<br />
6. Convolution (product)<br />
If 〈f0,f1,f2,...〉 ←→ F (x) <strong>and</strong> 〈g0,g1,g2,...〉 ←→ G(x), then<br />
〈h0,h1,h2,...〉 ←→ F (x) · G(x),<br />
where hn = f0gn + f1gn−1 + f2gn−2 + ··· + fng0.<br />
Proof.<br />
F (x) · G(x) = (f0 + f1x + f2x 2 + ...)(g0 + g1x + g2x 2 + ...)<br />
= f0g0 + (f0g1 + f1g0)x + (f0g2 + f1g1 + f2g0)x 2 + ...<br />
Q.E.D.
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