Taking the derivative of a generating function and trying to find the $n^{th}$ derivative

Question

When we are working with a generating function of a given sequence, when we take the derivative, we normally multiply by $x$ to shift the series back due to the derivative causing a shift in the opposite direction. Now, say we are trying to find the $n^{th}$ derivative of a generating function. Do we multiply by $x$ each time? I am trying to find a pattern.

Answer 1

To get the derivative, you differentiate. The $z \mathrm{D}$ trick you describe is for something different. Say you have: $$ A(z) = \sum_{n \ge 0} a_n z^n $$ If you compute the derivative: $$ A'(z) = \sum_{n \ge 1} n a_n z^{n - 1} = \sum_{n \ge 0} (n + 1) a_{n + 1} z^n $$ and again: $$ A''(z) = \sum_{n \ge 2} n (n - 1) a_n z^{n - 2} = \sum_{n \ge 0} (n + 2) (n + 1) a_{n + 2} z^n $$ To put back the $z$ "lost" by differentiation, you multiply by $z$, i.e., $$ z A'(z) = z \mathrm{D} A(z) = \sum_{n \ge 0} n a_n z^n $$ and if you are interested in the sequence $n^r a_n$, you get it by: $$ (z \mathrm{D})^r A(z) = \sum_{n \ge 0} n^r a_n z^n $$