The raising and lowering operators and for a sequence of functions , with and , defined by

and

have the commutator relation

with respect to action on the space spanned by this sequence of functions.

If for any particular natural number

,

then

,

implying

.

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 that also

.

An important application of the Pincherle derivative is to connecting different reps of the raising operators of Appell sequences:

The iconic ladder operators are and for the the powers , the prototypical Appell sequence of polynomials (see the post Bernoulli Appells for more on Appell sequences), so

,

and

.

If , then is the raising operator (see Bernoulli Appells) for an Appell sequence with moments given by the coefficients of the Taylor series for , i.e., ; lowering operator ; and e.g.f. .

**Related stuff**:

“The many avatars of a simple algebra” by Coutinho

http://mathoverflow.net/questions/97512/in-splendid-isolation/98213#98213

https://tcjpn.wordpress.com/2014/08/03/goin-with-the-flow-logarithm-of-the-derivative/

https://tcjpn.wordpress.com/2015/11/21/the-creation-raising-operators-for-appell-sequences/

https://tcjpn.wordpress.com/2014/12/10/appells-for-the-bernoullis/