Explicit Multi-point Taylor Polynomial

The multi-point Taylor polynomial, which is the general, unique and of minimum degree ($mk+m-1$) polynomial $P_{k,m}(x)$ which interpolates a function’s derivatives in multiple points is presented in its explicit form. A proof that this expression satisfies the multi-point Taylor polynomial’s defining property is given. Namely, it is proven that for a k-differentiable function $f$ and a set of different m-points $\{a_1,…,a_m\}$, this polynomial satisfies $P^{(n)}_{k,m}(a_i) = f^{(n)}(a_i)\quad \forall \, i = 1,…,m\quad \&\quad \forall \, n = 0,…,k$. A discussion regarding previous expressions presented in the literature, which mostly consisted in recursion formulas and not explicit formulas, is made.