I want to solve the following recurrence relation:
$$a_{n+1} = \alpha a_n + c_{n+1}$$
with initial conditions: $$ a_0 = 0$$ $$ c_1 = \beta$$
I actually know $c_{n+1}$ from the data in advance, however I am trying to find a solution to $a_{n+1}$ in terms of $c_{n+1}$. Is this possible?
You can just keep using the formula and get a pattern like this:
$a_{n+1} = \alpha a_n + c_{n+1} = \alpha(\alpha a_{n-1} + c_{n})+ c_{n+1} = \alpha(\alpha(\alpha a_{n-2} + c_{n-1})+c_n)+ c_{n+1} = \cdots = \alpha^{n+1}a_0 + \alpha^nc_1 + \alpha^{n-1}c_2+ \cdots \alpha c_n + c_{n+1}$
If you want in terms of cumulative sum
$a_{n+1} = \alpha a_n + c_{n+1}$
$a_{n} = \alpha a_{n-1} + c_{n}$
$a_{n-1} = \alpha a_{n-2} + c_{n-1}$
$\cdot \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot $
$\cdot \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot $
$\cdot \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot $
$a_2 = \alpha a_1 + c_2$
$a_1 = \alpha a_0 + c_1$
Adding
$a_{n+1} = (\alpha -1)\sum_{k=0}^n a_k + \sum_{k=1}^n c_k$