What kinds of expressions have a self-similar open form? A geometric series may be put in any of these 3 forms:


*

*Open form: $S = \sum^{i \in \mathbb{N_0}}\alpha^i$

*Self-similar form: $ S = 1 + \alpha S $

*Closed form: $ S = \frac{1}{1 - \alpha} $
If I take some other rational expression, say $\frac{1}{\alpha^2}$ or $\frac{\alpha + 1}{\alpha^2+1}$, would I be able to find a corresponding open form or self-similar form? What about other kinds of expressions, like polynomials and exponents?
 
P.S.   To clarify some:
An open form is not necessarily a sum. It may be a product, a continued fraction, or something else altogether. What matters is that there is not a fixed amount of operations.
A self-similar form is a recursive expression, that is, an expression defined in terms of itself. It may some time evaluate to a number; other times it will not evaluate at all, but remain formal.
 A: I don't know if this is the sort of thing you would be looking for.
Suppose we define the sequence $(q_n)_{n\in\Bbb N_0}$ by the recurrence
$$q_{n+2}=aq_{n+1}+bq_n\qquad n\ge0,$$
where $a$ and $b$ are non-zero constants. Assume the base cases $q_0,q_1$ are given. Say we define the function $$q(x)=\sum_{n\ge0}q_nx^n.$$
Then we can multiply both sides of our recurrence by $x^{n+2}$ and sum over $n\in\Bbb N_0$:
$$\sum_{n\ge0}q_{n+2}x^{n+2}=a\sum_{n\ge0}q_{n+1}x^{n+2}+b\sum_{n\ge0}q_n x^{n+2}.$$
The sum on the left is 
$$\sum_{n\ge0}q_{n+2}x^{n+2}=\sum_{n\ge2}q_nx^n=-q_0-q_1x+\sum_{n\ge0}q_nx^n=q(x)-q_1x-q_0.$$
The next sum is 
$$\sum_{n\ge0}q_{n+1}x^{n+2}=x\sum_{n\ge0}q_{n+1}x^{n+1}=x(q(x)-q_0).$$
And the last sum is $$\sum_{n\ge0}q_nx^{n+2}=x^2\sum_{n\ge0}q_nx^n=x^2 q(x).$$
Finally,
$$q(x)-q_1x-q_0=axq(x)-axq_0+bx^2q(x),$$
which is 
$$q(x)=\frac{(q_1-aq_0)x+q_0}{1-ax-bx^2}.$$
In a similar fashion, if the sequence $(q_n)$ is instead defined by 
$$q_{n+k}=\sum_{j=0}^{k-1}\alpha_j q_{n+j}$$
for constants $\alpha_j$ and $k$, then we can show that 
$$q(x)=\sum_{n\ge0}q_nx^n=\frac{\sum_{j=0}^{k-1}\left(q_jx^j-\alpha_j\sum_{r=0}^{j-1}q_rx^{k+r-j}\right)}{1-\sum_{j=0}^{k-1}\alpha_j x^{k-j}}.$$
