Given a generating function $F(x)=a+bx+cx^2+dx^3+\dotsb$, how do I truncate the $n$ lower order terms to get, for example if $n=2$: $cx^2+dx^3+\dotsb$?
For example, if I wanted to find $0a+1b+2c+\dotsb$, I would evaluate $$\left.\frac{dF(x)}{dx}\right|_{x=1}$$
This procedure can be used to find the expected value of a probability distribution given its generating function.
I want something similar for truncation of lower-order terms. This would give a cdf for a probability distribution. Since the cdf has a nice form for a binomial generating function, this suggests that there might be a nice way to arrive at it using generating function operators.
I vaguely remember learning this once, but flipping through the book generatingfunctionology didn't yield it.
If $f(x) = \sum\limits_{n=0}^\infty a_n x^n$, then $f''(x) = \sum\limits_{n=2}^\infty a_n n (n-1) x^{n-2}$, and $$ \int_0^x (x-t) f''(t)\ dt = \sum_{n=2}^\infty a_n x^n. $$ EDIT: More generally, $$ \int_0^x \dfrac{(x-t)^{k-1}}{(k-1)!} f^{(k)}(t)\ dt = \sum_{n=k}^\infty a_n x^n. $$