This is my first question on this site... yesterday I asked this on cs.so but they downwote and told me that so.cs is a research-level site, not for students... I hope it's appropriate to ask this question here ...
I'm looking for a general way to solve recursions with non-equal exponents on left and right. for example: $$ \mathrm{T}(n^a)=c\mathrm{T}(n^b)+\mathrm{P}_d(n)\;\;\;\; a \neq b $$
for example:
$ T(n) = 4T(\sqrt{n})+1 , \;\; (n>2) $
$ T(2)=1, \;\; (n \leq 2) $
Only answering the example.
Hint: What do you get, when you substitute $n=2^{2^k}$? Can you fill in the gaps? Try with small inputs $n=3,5,6,\ldots$ outside the set of values described in the first suggestion.