An interpolation formula of the derivatives of higher order (Q1804465)

In this paper the following interpolation formula is presented. For a function \(f(x) \in C^s [a,b]\) and \(n\) given points \(x_j\in [a,b]\), \(j = 1, \ldots, n\), \(x_i \neq x_j\), \(i \neq j\). Let the function values \(f(x_j)\) and derivative values \(f^{(j)} (x_i)\) \((i = 1, \ldots, n\); \(j = 1, \ldots, h_i - 1)\) be given. Then \(f(x) = F(x) + R_f(x)\), where \[ F(x) = \sum^n_{i = 1} \sum^{h_i - 1}_{j = 0} f^{(j)} (x_i) a_{ij} (x), \] \[ a_{ij} (x) = {Q(x) \over X_i^{h_i}} \sum^{h_i - 1 -j}_{k = 0} {1 \over j!k!} \left( {X_i^{h_i} \over Q(x)} \right)^{(k)}_{x = x_i} X_i^{j + k}, \] \[ f^{(0)} (x_i) : = f(x_i), \quad Q(x) = X_1^{h_1} \cdot X_2^{h_2} \cdots X_n^{k_n},\;X_i = (x - x_i),\;i = 1, \ldots, n. \] This is a well-known formula which may be found in monographies or literatures.