A general 1st order recurrence relation can be expressed as $X_{n+1} = f(X_{n},n)$.
I am looking for a source, list of functions or statement about $f$ such that an $F$ exists which expresses $X_{n} = F(f, X_0, n)$ as a closed form.
For example, a fairly simple function: $f(X_n,n) = b_{n} + a_n X_n$ can be rewritten as
\begin{eqnarray} X_{n+1} &=& b_{n} + a_n X_n \\ &=& b_{n} + a_n (b_{n-1} + a_{n-1} (b_{n-2} +\dots)) \\ &=& \left(\sum_{i=0}^{n} b_{n-i}\prod_{j>n-i}^{n}a_j \right) + X_0\prod_{j=0}^{n}a_j \\ &=& b_n + \left(\sum_{i=0}^{n-1} b_{i}\prod_{j=i+1}^{n}a_j \right) + X_0\prod_{j=0}^{n}a_j \\ &=& F(f,X_0,n+1) \end{eqnarray}
What can be said about $f$ if we require $F$'s existence?
By closed form I mean the a combination of all basic operations (+,-,*,/,^, $\sum, \prod \dots$), without the reference to a previously determined $X_n$
Edit for clarity: It seems function like \begin{equation}f(X_n,n) = a_n(1 - s_n^2) + s_n^2\frac{X_n + b_n}{2} + s_n\frac{|X_n - b_n|}{2} \end{equation} do not have such a closed form. It seems like this type of non-linearity already leads to a falsification of existence.