Applying the recursive function on numpy arrays will be faster than the current answer.
df = pd.DataFrame(np.repeat(np.arange(2, 6),3).reshape(4,3), columns=['A', 'B', 'D'])
new = [df.D.values[0]]
for i in range(1, len(df.index)):
new.append(new[i-1]*df.A.values[i]+df.B.values[i])
df['C'] = new
Output
A B D C
0 1 1 1 1
1 2 2 2 4
2 3 3 3 15
3 4 4 4 64
4 5 5 5 325