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>
$$
\boxed{N_{i,k}(t)=\frac{t-t_i}{t_{i+k-1}-t_i}N_{i,k-1}(t)
+\frac{t_{i+k}-t}{t_{i+k}-t_{i+1}}N_{i+1,k-1}(t)}.
$$

Solution

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

{c}

For $h=fg$,
$$
h[t_0,\ldots,t_k]=\sum_{m=0}^kf[t_0,\ldots,t_m]g[t_m,\ldots,t_k].
$$


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