Solving a peculiar recurence relation
In my opinion the problem statement seems poorly posed.  It's not clear what is meant by the problem statement.

Normally, if we write $a$ or $b$ the assumption is that they are a constant: they do not depend on $n$.  If they are intended to be a function of $n$, then they should be written as $a(n)$ or $b(n)$.  Since that wasn't done, the only assumption I can see that makes sense is to assume they are constants.  But then if they are constants, then writing $a,b = \omega(1)$ doesn't seem to me to make sense.

So, I suggest you go talk to your instructor to ask them what they had in mind and how one should interpret the question.  We can't solve the question until we know what the question is asking.

Basically, I am validating that the confusion you are having seems reasonable: it seems reasonable to me to be confused about that, as I find the problem statement confusing.