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 ,