I'm reading through the first chapters of "Introduction to the analysis of algorithm" by Sedgewick. I wasn't familiar with the use of generating functions, and complex analysis in general to analyse algorithms. I do understand the use of the tool is in practice simple (at least it seems to me in that way from what I read so far).
I still don't get though what's the particular advantage of using a generating function over standard techniques of difference equations when analysing some recurrence extrapolated by some algorithm.
What's the benefit of using generating functions?
I can also see "analytic combinatoric" as topic, I haven't read through that yet, what's the advantage of that formalism instead? Is there some case where analysing in that way is really easy while using standard technique isn't?