I have the following recursive function that takes three forms and I want to express it in one form:
Initial: $f(x) = m * f(x-1)$, $f(0) = value.$
Forms:
1 - $f(t) = m * f(t-1)$. where t is at $+\infty$. So I want to start from the basic value at $t=0$ and then keep applying this to infinity. i.e. I want $f(x)$ where $x=\infty$.
2 - $f(t) = m * f(t-1)$. where $t >= 0$. I want to apply this starting from the basic value at $t=0$ and end at some value $t=k$ that I specify. So $f(x)$ where $x = k$.
3- $f(t) = \sum_{i}^{t}{m*f(i)}$. Here the function gets evaluated as $mf(1) + mf(2) +...+mf(t)$.
What I'm looking for is to express the three forms into one form that I can just manipulate it by changing parameters.
This is how I'm trying to do it, but I'm not sure if my mathematical expression is correct.
Compact form: $f(t) = \sum_{i}^{k}{m*f(i)}$
Form 1: $t = +\infty, k=1, i=+\infty$. This is definitely wrong I guess because $i$ can't start at $+\infty$ and end at $1$.
Form 2: $t = c, k=1, i=c$. This has the same issue as the previous step. I just want to evaluate the function at $t=c$ without any summation.
Form 3: $t=k$.
As you can see I have a problem to translate the forms into one compact form. I hope you can help.