Let us give  the power series

f(x) = c0 + c1 x + c2 xx  + c3 xxx +....

From this define a modified series with capital F

F(x) = c0 + c1 x  +c2  x(x+h)  + c3 x(x+h)(x+2h) +...

This is really a function of two variables, but concéntrate on the x variable.

The derivative of the original series is Df

Df = c1 + 2 c2 x +...

We call now the transformation of the derivative series  TDF

The point of interest is now

F(x+h) - F(x) = h TDF(x+h)

We get the right hand side by first calculating the transformation of the derivative function Df, to get TDF and then this series

is evaluated not at x but at x+h.

These operations are not  commutative.

This shows the connection between the continium and the discrete in

concrete form. Hopefully the result is correct.

If h tends to zero one just gets Df=Df

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