Changes

And of course, all the higher order differences are zero.

And of course, all the higher order differences are zero.

This leads to thinking of <math>X\!</math> as having an extended state <math>(x, dx, d^2 x, \ldots, d^k x),</math> and this additional language gives us the facility of describing state transitions in terms of the various orders of differences.  For example, the rule <math>x' = (x)\!</math> can now be expressed by the rule <math>dx = 1.\!</math>

This leads to thinking of <math>X\!</math> as having an extended state <math>(x, dx, d^2 x, \ldots, d^k x),</math> and this additional language gives us the facility of describing state transitions in terms of the various orders of differences.  For example, the rule <math>\texttt{x' = (x)}</math> can now be expressed by the rule <math>\texttt{dx = 1}.</math>

I'll leave you to muse on the possibilities of that.

I'll leave you to muse on the possibilities of that.