Cox-de Boor formula:
\[ B_{i,0} := \left\{ \begin{array}{ll} 1\ \ \text{if}\ t_i \leq x \le t_{i+1} \\ 0\ \ \text{otherwise.} \end{array} \right. \] \[ B_{i,k} := \frac{x-t_i}{t_{i+k} - t_i} B_{i,k-1}(x) + \frac{t_{i+k+1} - x}{t_{i+k+1} - t_{i+1}} B_{i+1,k-1}(x) \]
\[ \frac{dB_{i,k}(x)}{dx} = k \left( \frac{B_{i,k-1}(x)}{t_{i+k} - t_i} - \frac{B_{i+1,k-1}(x)}{t_{i+k+1} - t_{i+1}} \right) \]
This implies that
\[ \frac{d}{dx}\sum_{i} \alpha_i B_{i,k} = \sum_{i=r-k+2}^{s-1} k \frac{\alpha_i - \alpha_{i-1}}{t_{i+k} - t_i} B_{i,k-1} \ \ \text{on}\ \ [t_r, t_s] \]
Let \[ \beta_i = k \frac{\alpha_i - \alpha_{i-1}}{t_{i+k} - t_i} \] Then \[ \int_0^\zeta \frac{d}{dx}\sum_{i} \alpha_i B_{i,k} \ dx = \int_0^\zeta \sum_{i=r-k+2}^{s-1} \beta_i B_{i,k-1} \ dx \] \[ \sum_{i} \alpha_i B_{i,k} \mid_0^\zeta = \int_0^\zeta \sum_{i=r-k+2}^{s-1} \beta_i B_{i,k-1} \ dx \] If \(\beta_i\) is considered as known and we have to find \(\alpha_i\), we get \[ \alpha_i = \beta_i \frac{t_{i+k}-t_i}{k} + \alpha_{i-1} \] here \(\alpha_0\) being a freely defined constant. In term of cumulative sum, it gives \[ \alpha_j = \sum_{i=0}^j \beta_i \frac{t_{i+k}-t_i}{k} \]