Finding the closed form of recurrent sequences What are the famous (general) methods to find the closed form of a given recurrent sequence?
The only method I know of is the "generating function" method. However it only works in very special cases; the given recurrence should look like (where the recurrent sequence is $U_n$): 
$$\sum_{k=0}^m \alpha_k U_{n - k} = W_n$$
Where $(W_n)$ is nice enough. 
But what about the cases, say: $f(U_n) =W_n$?
I know there is no general procedure to solve a recurrence which may be of any type, but:

I would like to know: what are some famous techniques, special cases, famous books, etc. that handle this subject?


Wilf's generatingfunctionology is considered to be THE book on https://www2.math.upenn.edu/~wilf/DownldGF.html
You might also want to check out A=B by Petkovšek, Wilf, Zeilberger: https://www2.math.upenn.edu/~wilf/AeqB.html
The general case gets VERY messy. Integer partitions are an example where there is a "nice" recurrence, but no "short" formula is known.
