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 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
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. .
“The many avatars of a simple algebra” by Coutinho