In "Four examples of Beilinson-Bernstein localization", Anna Romanov introduces the basis
$m_k = \frac{(-1)^k}{k!} \partial^k \delta $
on p. 9, where $\partial$ is a partial derivative and $\delta$ is the 'Dirac indicator function'. This is related to the Dirac-Appell sequences I wrote about quite some time ago. In fact, the identity just below (4.12) on p. 6 morphs into
$\psi(\partial_z) = w^{-(t-1)}\partial_ww^{t-1} = (1+u)^{-(t-1)}\partial_u(1+u)^{t-1}$
for $w = u+1$, which is an instance, with
$A(u)=(1+u)^{-(t-1)},$
of the generic raising operator $R$ for a generic Dirac-Appell sequence $A_n(u)$, characterized by
$A_n(u) = A(u) \partial_u^n \delta(u) = A(u) \; \partial_u \; (A(u))^{-1} A_{n-1}(u) = R A_{n-1}(u).$
(See also the casual intro "A tour via examples of the Beilinson-Bernstein localizations" by Romanov.)
Can someone recommend papers that present arguments of Romanov from the perspective of the Dirac-Appell sequences, essentially Laplace-transformed Sheffer-Appell sequences, or the conventional Sheffer formalism--in a sense, expository papers that span the nether land between the concepts and terminology of algebraic and differential geometry / topology and category theory applied to the infinite dimensional representation theory of $SL_2$, or $SL_n$, and the concepts and terminology of the combinatorics and calculus of the Sheffer-Appell formalism or Dirac-Appell?
Researchers of the caliber of Hirzebruch were once constantly doing such, knowingly or unknowingly, for the Sheffer formalism, see, e.g., the related "Formal Groups, Witt vectors and Free Probability" by Roland Friedrich and John McKay and A263916. Feinsilver and his collaborators have addressed similar points, e.g., in "Lie algebras, Representations, and Analytic Semigroups through Dual Vector Fields". For infinite dimensional matrix and diff op reps, based on raising and lowering ops, of $\frak{sl}_n(\mathbb{C})$ and relations to the Witt and Virasoro algebras and Sheffer formalism, see my sketch "Infinigens, the Pascal Triangle, and the Witt and Virasoro Algebras".
Edit: Mar. 13, 2023
On the significance of the conjugating factors:
$F(x) = (1+x)^{a} = 1 + a x + (a - 1) (a) \frac{x^2}{2!} + (a - 1)(a - 2) (a) \frac{x^3}{3!} + \cdots$
is the generating series for the falling factorial polynomials $n!\binom{a}{n}$,
$R(x) = (1+x)^{-a} = 1 - a x + (a + 1)(a) \frac{ x^2}{2!} - (a + 2) (a + 1)(a) \frac{x^3}{3!} + \cdots$
is the e.g.f. for the rising factorial polynomials $(-1)^n n!\binom{a+n-1}{n}$.
These polynomials are significant in diverse fields of mathematics, as the related OEIS reveal.
The two are related by the identity $\binom{x}{n} = (-1)^n \binom{-x-1+n}{n}$ and this is related to an exceptional combinatorial reciprocity (CR) discussed in this MO-Q and the links therein.
The rising factorial, Pochhammer symbol, or Stirling polynomials of the first kind are operationally intimately related to fractional calculus and operational definitions of Kummer's confluent hypergeometric functions, or generalized Laguerre polynomials / functions, THE special functions for much of classical and quantum physics involving vibrations or oscillatory phenomena.
The lowering factorial is also of operational importance. In fact, it is the umbral inverse of the revered Bell / Stirling polynomials of the second kind, operationally defined by the normal ordering of the state or Euler operator $(xD_x)^n$ (see, e.g., this MSE-A), and a core construct in combinatorics, generalizing to the Faa di Bruno composition / refined Stirling polynomials of the second kind of A036040.
The falling factorial polynomials of A008275 / A048994 / A094638, or Stirling polynomials of the first kind, are core constructs in combinatorics, operational calculus, and analysis as are their refinement the cycle index polynomials of the symmetric groups $S_n$ of A036039. Note the relation to $m$-ary trees and the generalized Witt algebra demonstrated in A094638].
The normalized Laguerre polynomials and the closely linked Lah polynomials (normalized Laguerre polynomials of order -1 and, CR again, closely related to the normalized Laguerre polynomials of order 1) fit in here in other ways:
First, see my comment to my answer to this MO-Q "the rising factorials are the coefficients of the Laguerre series expansion of the Heaviside step function and its derivatives, the derivatives of the Dirac delta function, while the falling factorials are the Laguerre expansion of the divided powers \frac{x^n}{n!}" as noted in my answer to this MO-Q.
Second, the Narayana triangle A001263, one reduction of the noncrossing partition polynomials of A134264--which can be defined via compositional inversion, is intimately related to the coefficients of the Lah polynomials (search on Lah at A001263).
Third, the Lah polynomials umbrally transform the falling factorial polynomials into the rising factorial polynomials and vice versa (see my MO-answer).
Fourth, the refined multivariate Lah partition polynomials of A130561 can be umbrally transformed into the associahedra Lagrange inversion polynomials of normalized A133437 using the regular single variable Lah polynomials (see my post "Lagrange a la Lah").
Fifth: The operator manifestations of the Laguerre and Lah polynomials are related to modular functions and $sl_2$ and $sl_n$, which are related to the falling factorials via the generalized Witt vectors $(x^{y+1} D_x)^n$ (see A094638 again).
(This is not a comprehensive list.)
