# How to define this recursive function?

Consider the relation

$$t_{k}(n) = 2 t_{k-1}(frac{n}{2}) + f(n)$$

Where $$t_{0}(n) = x^p$$ and $$f(n)=alpha n^q$$.

How can I get an arbitrary element $$t_{k}(n)$$ for example with $$p=2,q=1,alpha=1$$ using Mathematica syntax?