In "Left differential operators on non-commutative algebras" on p. 4, Michiel Hazewinkel displays "precisely the right definition of differential operator" as
$$D\; X^n = F(\tfrac{d}{dX})\; X^n = (f_0 + f_1 \tfrac{d}{dX} + f_2 \tfrac{d^2}{dX^2} + \dotsb + f_n \tfrac{d^n}{dX^n})\;(X^n).$$
I'm most familiar with the cases in which $f_0=1$ and otherwise $f_n$ is independent of $x$, as occurs in the Sheffer-Appell calculus; a variation in which $F_m(\tfrac{d}{dx})= (g(x)\tfrac{d}{dx})^m$, as occurs in the Sheffer-binomial calculus, the diff op calculus of compositional inversion and formal group laws, and Witt algebra (see (MO-Q1, MO-Q2, MO-Q3, MP-Q4, MO-Q5, MO-6 ); and another variation in which operators are of the Rodriguez-type as in this MO-Q and this one or of the raising diff ops of general Sheffer sequences (see eqn. 9 of Wolfram MathWorld's Sheffer sequence and also eqns. 4, 5, 7, 12, and 16, all with $t$ replaced by $\tfrac{d}{dx}$). See also this MSE-Q1 and MSE-Q2.
(Update July 6, 2023: My pdf at my blog post "The umbral compositional inverse and lowering and raising operators of a Sheffer polynomial sequence" gives the diff ops for the substitution and ladder ops--the raising / creation op $R$ and the lowering / destruction / annihilation op $L$--for general Sheffer sequences, which include the Appell and binomial subgroups. This MSE answer has additional info on and links to more general related diff ops.)
I'll sketch only the Appell case from the perspective of Graves and Pincherle, but Carmichael, in "Linear differential equations of infinite order" (1935) gave a survey of the more general situation involving Hazewinkel's diff op, sketching the work of "two pioneers" in the subject ('transmutations')--Pincherle and Bourlet--and later researchers.
With $f_0 =1$ and $f_n = \frac{\bar{f}_n}{n!}$ are independent of $x$, this is precisely the diff op generator of Appell–Sheffer sequences $A_n(\bar{f}_1,\bar{f}_2,\dotsc,\bar{f}_n;x)$, or in umbral notation and suppressing the dependence on $\bar{f}_k$, the moments, $(A.(x))^n = A_n(x)$, defined by the e.g.f.
$$\text{$e^{A.(x)t} = A(t) e^{xt}$ with $A(t) = e^{\bar{f}.t}$.}$$
Then
$$A(\tfrac{d}{dx}) x^n = A_n(x).$$
Hazewinkel states if $D$ is a diff op of order $\leq n$ on a commutative algebra $A$ over $k$ and $l_x$ denotes a $k$-morphism of multiplication by $x$, then the commutator, or adjoint,
$$[D,l_x] = Dl_x - l_xD = \operatorname{ad}(x)D$$
is a diff op of order $\leq (n-1)$.
For Appell sequences, with $h(z)$ analytic at $z$ and $L$ and $R$ lowering and raising operators defined by, e.g.,
$$L \; A_n(x) =n \; A_{n-1}(x)$$
and
$$R \; A_n(x) = A_{n+1}(x)$$
this is an analogue of the Graves–Pincherle-Lie derivative
$$[h(L),R] = h'(D) = \tfrac{d}{dL}h(L),$$
touted by Rota in his (and that of colleagues Roman, Taylor, Joni, et al.) version of the umbral-Sheffer calculus, the finite operator calculus, and central to the calculus of Lyndon words and normal ordering of a class of diff ops (think quantum mechanics). The prototypical (identity) Appell and binomial Sheffer sequence is $P_n(x) = x^n$ with $R=x$ and $L= \frac{d}{dx}$. A Graves–Pincherle derivative also holds for $L$ and $R$ interchanged.
For any strict Appell sequence, i.e., with $A_0(0) =1$, $L =\frac{d}{dx}$ and $R$ also has the form of $A(\frac{d}{dx})$; precisely, it's the conjugation $R = A(\frac{d}{dx}) \; x \;(A(\frac{d}{dx}))^{-1}$. The umbral inverse Appell sequence $\hat{A}_n(x)$ such that $A_n(\hat{A}.(x))=x^n$ has the generator $(A(\frac{d}{dx}))^{-1} = 1/A(\frac{d}{dx})$.
The diff op calculus of Sheffer sequences all translates into a group algebra of lower triangular matrices (a family of Riordan arrays), intimately related to symmetric functions and multiplicative and compositional inversions, which dovetails nicely with Hazelwinkel's and others' ultimate interest in noncommutative symmetric functions. A similar group algebra cast in matrix form with inverse matrix pairs for noncommutative symmetric functions is presented by Novelli and Thibon in "Noncommutative Symmetric Functions and Lagrange Inversion II: Noncrossing partitions and the Farahat-Higman algebra".
Hazewinkel goes on to elaborate on the diff op and writes, "The material treated in sections 2 – 6 above is merely the affine case of what is in [6 Grothendieck et al.], §16 in greater depth and in the greater generality of schemes (and without examples)."
Perhaps this brief intro to the calculus of Appell sequences will spark some neural connections for those more familiar with schemes (and / or symmetric functions) than Appell sequences.
My question:
What literature is available that explicitly addresses the parallelism between the Sheffer diff op calculus and the more general formalism of the affine case of Grothendieck schemes Hazewinkel references?
This has bearing on this MO-Q.
