The raising and lowering operators and for a sequence of functions , with and , defined by
have the commutator relation
with respect to action on the space spanned by this sequence of functions.
If for any particular natural number
Since this holds for , the relation holds for all natural numbers, and formally for a function analytic about the origin (or a formal power series or exponential generating function)
The reader should be able to modify the argument to show the dual relation