- Say I have a generating function $f(z) = \sum_{n=0}^\infty a_nz^n$. And I want a generating function for $g(z) = \sum_{n=0}^\infty n!a_nz^n$ for the same $a_n$. How do I modify $f(z)$?
EDIT: In my case $f(z) = \frac{1}{(1-z)(2-e^z)}$.
- Are there any results on evaluating coefficients of generating functions modulo primes?
Thanks.
Regarding 1, we have (this is the Borel transform):
$$g(z) = \sum_{n=0}^\infty n!a_nz^n$$
$$g(z) = \sum_{n=0}^\infty n!a_n \frac{1}{n!}\left(\int_0^\infty e^{-t}t^ndt\right)z^n$$
$$g(z) = \int_0^\infty e^{-t}\sum_{n=0}^\infty a_n (tz)^ndt$$ $$g(z) = \int_0^\infty e^{-t}f(tz)dt$$
So in your case:
$$g(z) = \int_0^\infty \frac{e^{-t}}{(1-tz)(2-e^{tz})}dt$$