## 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

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

then

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

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

implying

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

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

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

and

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

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

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

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.

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

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/2012/11/29/infinigens-the-pascal-pyramid-and-the-witt-and-virasoro-algebras/

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/

http://oeis.org/A263634