Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 3 / b / Solution

Apply the Leibniz rule for divided differences to $(x-t{)}_{+}^{k-1}=(x-t)(x-t{)}_{+}^{k-2}$. Only the zeroth and first divided differences of the linear factor survive. After applying the normalization, this gives the Cox-de Boor recursion formula