The Pincherle Derivative and the Appell Raising Operator

The raising and lowering operators R and L for a sequence of functions \psi_n(x), with n= 0,1, 2, ... and \psi_0(x)=1, defined by

R \; \psi_n(x) = \psi_{n+1}(x) and L \; \psi_n(x) = n \; \psi_{n-1}(x)

have the commutator relation

[L,R] = LR-RL = 1

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

If for m any particular natural number

[L^m,R] = m \; L^{m-1} = \frac{d}{dL}L^m ,


mL^m =L \; [L^m,R] = L^{m+1}R - LRL^{m}

= L^{m+1}R - (1+RL)L^{m} = L^{m+1}R - RL^{m+1} - L^m ,


[L^{m+1},R] = (m+1) L^{m} = \frac{d}{dL}L^{m+1}.

Since this holds for m=1, the relation holds for all natural numbers, and formally for a function f(x)=e^{a.x} analytic about the origin (or a formal power series or exponential  generating function)

[f(L),R] = [e^{a.L},R]= \frac{d}{dL}e^{a.L} = a. \; e^{a.L}=\frac{d}{dL}f(L).

The reader should be able to modify the argument to show the dual relation

[L,f (R)] = \frac {d}{dR}f(R) .

This should be expected from representing the iconic operators in a dual Fourier space through which multiplication by x  becomes proportional to a derivation in the Fourier reciprocal space by, say d/dp, and derivation d/dx, to multiplication by p.

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

The iconic ladder operators are L_p = D = d/dx and R_p=x for the the powers x^n, the prototypical Appell sequence of polynomials (see the post Bernoulli Appells for more on Appell sequences), so

[f(D),x] = [f(L_p),R_p]=\frac {d}{dD}f(D) = f'(D),


f(D)^{-1}  \; x  \; f(D) =x-x+f(D)^{-1} \; x \;  f(D)

=  x - f(D)^{-1} \; [f(D),x] = x - f(D)^{-1}f'(D) = x-\frac {d}{dD} \; \ln(f(D))

= \frac{d}{dt}|_{t=D}  \; \ln[e^{x \; t} / f(t)] = \frac{d}{dt}|_{t=D}  \; \ln[e^{x \; t} e^{c.t}]=\frac{d}{dt}|_{t=D}  \; \ln[e^{(c.+ x) \; t}].

If f(0) = 1, then R_A = f(D)^{-1}  \; x  \; f(D)=  x- \frac{d}{dD} \; \ln(f(D)) is the raising operator (see Bernoulli Appells) for an Appell sequence p_n(x) with moments given by the coefficients (c.)^n = c_n of the Taylor series for f(x)^{-1}, i.e., (c. +x)^n = p_n(x);  lowering operator L_A = D; and  e.g.f. e^{tp.(x)} = e^{x \; t} / f(t)= e^{(c. + x)t}.

Note that \frac{1}{f(D)} e^{xt} = \frac{1}{f(t)}e^{xt}=e^{p.(x)t}, so \frac{1}{f(D)}x^n = p_n(x).

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 R_A = R + h(L) and L_A = L, where h(x) is a Taylor / power series about the origin or a formal exponential / ordinary generating function.

Added Oct. 2, 2016:

The Pincherle derivative [T^n(L),R] = \frac {d}{dL}T^n(L)  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 \delta^{(n)}(x) with L = -x and R = D = d/dx and e.g.f. e^{tD} \delta(x) = \delta(x+t). Another example is provided by OEIS A099174 with the D-A sequence H_n(x) = h_n(D) \delta(x) where h_n(x) are the modified Hermite polynomials listed in the Example section of the entry.  The lowering and raising operators of this D-A sequence are L = -x and R_{DA} = -x + D,  and the e.g.f.  is \exp[t H.(x)] = e^{t^2/2} e^{t D} \delta(x) = e^{t^2/2} \delta(x+t).

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.


Related stuff:

The many avatars of a simple algebra” by Coutinho


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