Is it possible to convert a polynomial into a recurrence relation? If so, how? I have been trying to do this for quite a while, but generally speaking the partially relevant information I could find on the internet only dealt with the question: "How does on convert a recurrence relation into... well, a non-recurrence relation?" 
Let's start of with a simple example: $f(n) = 34n^3+51n^2+27n+5$. How do we find $f_{n}$? 
I'd really like to see this solved in analogy with the following: Consider $g(n)=n^6$ We can then find the recursion formula: $g_n=((g_{n-1})^{1/6}+1)^6$. What about $f_{n}$?
We could generalize this question in a number of ways. For instance, is it (also) possible too turn an infinite polynomial, like the Taylor Series expansion of a trigonometric formula, into a recursion formula? Furthermore, what happens when we allow the coefficients of the polynomial to be real and even complex?
Thanks,
Max
Bonus side-question: How, if at all, are "generating functions" useful in this context?

Expand $f(n+1)$ and you'll get $f(n+1)=f(n)+g(n)$ for some $g$. For $f(n)=34n^3+51n^2+27n+5$, you get $f(n+1)=f(n)+102n^2+204n+112$. 
