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?