Newton's theorem with respect to a lot of centers and their applications (Q1367373)

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