Let's say the initial conditions of a second order recurrence relation are given for $v_{-1}$ and $v_{0}$ rather than the classical $v_{0}$ and $u_{1}$. It seems that the usual formula for the homogeneous solution $v_{n} = \alpha r_{1}^{n} + \beta r_{2}^{n}$ does not work anymore.
From what I can see, it seems that we get $v_{n-1} = \alpha r_{1}^{n} + \beta r_{2}^{n}$ instead.
Actually, why is the usual formula only working for an initial term given with $n=0$ in the first place? Also, how does the change of formula (for other cases) occur exactly?
I could not find anywhere a general formula including cases where the initial term is giving for $n$ different from $0$, what would it be?
Just in case an example is needed (the following can be skipped otherwise), here is a situation where this phenomenon occur (this example is not from me, I will just describe it as I read it in my book): let's say we have a real world situation corresponding to the reccurence relation $v_{n+2} = (q-1)(v_{n+1} + v_{n})$ where $n \geq 1$ and $q \geq 2$ and with initial conditions $v_{1} = q$ and $v_{2} = q^2$. Then we say we can start the recurrence earlier by defining $v_{0} = q/(q-1)$ and $v_{-1} = 0$. (I guess it is just a mathematical extension in order to make the calculations of the constants of the solution easier, but tell me if it could be for some other reason). From there, it seems that the form of the solution is as described above: $v_{n-1} = \alpha r_{1}^{n} + \beta r_{2}^{n}$.
$$\alpha r_1^n+ \beta r_2^n=(\alpha r_1)r_1^{n-1}+(\beta r_2)r_2^{n-1}.$$