Editing Divided differences (section)

===Polynomials and power series===
The matrix 
<math display="block">
J =
\begin{pmatrix}
x_0 & 1 & 0 & 0 & \cdots & 0 \\
0 & x_1 & 1 & 0 & \cdots & 0 \\
0 & 0 & x_2 & 1 &        & 0 \\
\vdots & \vdots & & \ddots & \ddots & \\
0 & 0 & 0 & 0 &    \; \ddots     & 1\\
0 & 0 & 0 & 0 &          & x_n
\end{pmatrix}
</math>
contains the divided difference scheme for the [[identity function]] with respect to the nodes <math>x_0,\dots,x_n</math>, thus <math>J^m</math> contains the divided differences for the [[monomial|power function]] with [[exponent]] <math>m</math>.
Consequently, you can obtain the divided differences for a [[polynomial function]] <math>p</math> by applying <math>p</math> to the matrix <math>J</math>: If
<math display="block">p(\xi) = a_0 + a_1 \cdot \xi + \dots + a_m \cdot \xi^m</math>
and
<math display="block">p(J) = a_0 + a_1\cdot J + \dots + a_m\cdot J^m</math>
then
<math display="block">T_p(x) = p(J).</math>
This is known as ''Opitz' formula''.<ref>[[Carl de Boor|de Boor, Carl]], ''Divided Differences'', Surv. Approx. Theory  1  (2005), 46–69, [http://www.emis.de/journals/SAT/papers/2/]</ref><ref>Opitz, G. ''Steigungsmatrizen'', Z. Angew. Math. Mech. (1964), 44, T52–T54</ref>

Now consider increasing the degree of <math>p</math> to infinity, i.e. turn the Taylor polynomial into a [[Taylor series]].
Let <math>f</math> be a function which corresponds to a [[power series]].
You can compute the divided difference scheme for <math>f</math> by applying the corresponding matrix series to <math>J</math>:
If
<math display="block">f(\xi) = \sum_{k=0}^\infty a_k \xi^k</math>
and
<math display="block">f(J)=\sum_{k=0}^\infty a_k J^k</math>
then
<math display="block">T_f(x)=f(J).</math>