Consider the following difference equation $$y_{n}=-\sum_{k=1}^{q}a_{k}y_{n-k}+\sum_{m=0}^{p}b_{m}x_{n-m}$$
I know that this is supposed to be LTI iff $y_{-q}=y_{-q+1}=\cdots=y_{-1}=0$. How does one go about showing this. I have tried but I don't really see where I need the initial conditions. I can show it is linear but I am struggling with the time invariance.
Applying $B$ to $x_{n}$ I get $x_{n-1}$. Then using the system above I get
$$y_{n-1}=-\sum_{k=1}^{q}a_{k}y_{n-k-1}+\sum_{m=0}^{p}b_{m}x_{n-m-1}$$
Applying the system first gets me $y_{n}$ as shown at the start of the question. Applying $B$ then gives the same thing.
Why do I need the initial conditions?