Convolution

$latex (2^0 +2^1 + 2^2 +…). (3^0 +3^1 + 3^2 +…)
$
$latex = (2^{0}3^{0})+ (2^{0}3^{1}+ 2^{1} 3^{0}) + (2^{0} 3^{2} + 2^{1} 3^{1} + 2^{2} 3^{0} ) + …
$
$latex displaystyle
= sum_{n=0}^{infty}
sum_{k=0}^{n}
2^{k} 3^{n-k}
$

Let the sequence $latex left { a_{n} right }$ convolved with another sequence $latex left { b_{n} right }$

$latex boxed {
left { a_{n} right } = left { a_{0}, a_{1}, a_{2}, …, a_{n}, … right }
}$
Its correspondence $latex leftrightarrow $ the generating function:
$latex displaystyle boxed {
a(x) = sum_{k=0}^{n}a_{k}x^{k}
}$

$latex boxed {
left { b_{n} right } = left { b_{0}, b_{1}, b_{2}, …, b_{n}, … right }
}$
Its correspondence $latex leftrightarrow $ the generating function:
$latex displaystyle boxed {
b(x) = sum_{k=0}^{n}b_{k}x^{k}
}$

The convolution is $latex displaystyle boxed { left { a_{n}* b_{n} right } =
left { sum_{k=0}^{n} a_{k}b_{n-k}right }
&fg=aa0000&s=1}$