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
This should be expected from representing the iconic operators in a dual Fourier space through which multiplication by becomes proportional to a derivation in the Fourier reciprocal space by, say , and derivation , to multiplication by .
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. .
Note that , so .
We’ve been using the power basis, but one should be able to construct a generalized Appell sequence from the raising and lowering ops of any sequence by letting and , where is a Taylor / power series about the origin or a formal exponential / ordinary generating function.
Added Oct. 2, 2016:
The Pincherle derivative is being implicitly used in Eqn. 2.19 page 13 of “Mastering the master field” by Gopakumar and Gross. The raising and creation operators in the paper are analogous to those for a Laplace-dual Appell sequence, or Dirac-Appell sequence, comprised of the Dirac delta function and its derivatives, formed by taking the inverse Laplace transform of the polynomials of an Appell polynomial sequence. The basic D-A sequence can be defined as the sequence with and and e.g.f. . Another example is provided by OEIS A099174 with the D-A sequence where are the modified Hermite polynomials listed in the Example section of the entry. The lowering and raising operators of this D-A sequence are and , and the e.g.f. is .
Multiplication of an Appell polynomial by another corresponds to convolution integrals of the corresponding D-A functions, which have representations as Cauchy complex-contour-integral operators and band-limited Fourier transforms.
“The many avatars of a simple algebra” by Coutinho