I encountered the following sum:
$$\frac{c_{mn-p}}{c_0} = \sum_{d|p}\ \frac{a_{n-d}b_{m-\frac{p}{d}}}{a_0b_0} $$
Where
$$a_i=0 \text{ when } i<0 \text{ and } b_j=0\text{ when } j<0\text{ and n,m are positive integers } $$
I wish to write $a_i$ terms of $b$ and $c$ for all $i$, but I having trouble isolating $a$. I have tried deriving a recursion formula for $c_p$ in terms of $c$ to isolate either $a$ or $b$ but am having no luck. Will Mobius inversion help? Are there any pre-existing formulas in mathematics where this form is present? Any ideas?
Mobius inversion is the way to go. As long as either a or b is an arithmetic function with a Dirichlet inverse.