Generating Function

Example

Lemma

• If $f(x)$ is the generating function for

$$ a_0, a_1, a_2 $$

    then $f'(x)$ is the generating function for 

$$ a_1, 2a_2, 3a_3... $$

• Equivalently, $f'(x)$ is the generating function for

$$ {(i + 1)a_{i + 1}} $$

   or $xf'(x)$ is the generating function for 

$$ {ia_i} $$

• Example

Lemma

shift operator in signal processing or operator E in difference equations