I am reading about Newton's divided differences and I am confused by the following derivation of the coefficients of the Newton's polynomial.
The Newton's polynomial is given as
$P_n(x) = a_0 + a_1(x - x_0) + a_2(x-x_0)(x-x_1) + \dots + a_n(x-x_0)...(x-x_{n-1})$ (1)
A polynomial of order $n$ can be constructed from two polynomials of order $n-1$ by dividing the support in two parts like this
$P_{k,k+1,\dots,k+j}(x) = \dfrac{(x- x_k)P_{k+1,\dots,k+j}(x)-(x- x_{k+j})P_{k,\dots,k+j-1}(x)}{x_{k+j} - x_k}$ (2)
This I understand.
The next statement is confusing me.
Because a polynomial of order $n$ is unique for a specific support, an immediate consequence of this is that the coefficients $a_i$ in (1) are Newton's divided differences, because $a_j$ is the highest coefficient of a polynomial $P^*_{0,\dots,j}(x)$ for the $(j+1)$ support points $((x_0, y_0), \dots (x_j, y_j))$.
This "immediate consequence" is not at all immediate to me. :)
I understand that the divided differences can be derived in different ways for the n-th coefficient, there are answers already available for this question, I don't see how the interpolation with sub-polynomials immediately leads to $a_n = [y_0, \dots, y_n]$.