The order finite difference of function is defined by

From this definition, we have

and,

as well as

The function shown below generates (see Fig. 1).

delta_(g, n) := block(
local(f),
define(f[1](x),
g(x+1)-g(x)),
for i : 2 thru n do (
define(f[i](x),
f[i-1](x+1)-f[i-1](x))
),
return(f[n])
);

Fig. 1

Compare to the result of expanding (see Fig. 2)

Fig. 2

It *seems* that

Lets *prove* it!

We have already shown that (1) is true for .

Assuming (1) is true when :

When ,

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