The recursive part of this problem is short -- happening only in the mergesort method, but driving the entire algorithm.
Your goal is to make mergesort work, using recursion.
The split_array method is not mandatory to use, but I think it makes the solution a little easier to read and follow by making the mergesort method read almost exactly like the psuedocode, hiding all the details of how it works in other methods.
The hardest part of this problem, I think, will be writing the combine method and dealing with all the cases that occur when combining two lists of sorted numbers. This is an iterative solution, not a recursive solution.