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In a 2013 article, Cheng, Duncan and Harvey introduce the concept of umbral moonshine as a generalization of monstrous moonshine. The terminology they use, starting with the title, is common in umbral calculus of Rota, Roman and others. The articles by Cheng et al. lack any clarification on the homonymy, or links to works on umbral calculus, which suggests that the similarity in terminology may be coincidental. At the other hand, q-calculus emerges in texts on umbral calculus here and there, and a 2009 PhD thesis by Robinson loosely connects umbral calculus to the theory of vertex operator algebras, so it is not impossible to imagine that a common ground for umbral moonshine and umbral calculus could exist. Since the article treats pretty complex material I am not familiar with, I wasn't able to immediately understand from the text, if any analogy to the classical umbral calculus is present, so, before dedicating a substantial amount of time to further attempts, I would like to ask if anyone else already knows the answer.

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We used the word "umbral" because umbral moonshine involves mock modular forms whose lack of modularity is characterized by another function known as the shadow of the mock modular form. Umbra is the Latin word for shadow. We did not mean to imply any connection to umbral calculus.

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Despite the authors not intending to imply a link to umbral calculus per se, there is indeed a connection of the umbral calculus / Sheffer polynomial operator calculus to monstrous moonshine as I noted in my MO-Q "Combinatorics for the action of Virasoro / Kac–Schwarz operators: partition polynomials of free probability theory":

The association with monstrous moonshine is via the noncrossing partition polynomials of OEIS A134264, which are related to the compositional inversion (CI, or series reversion) of Laurent series with the free cumulant partition polynomials of free probablility theory as numerators and also the CI of o.g.f.s / formal power series via their shifted reciprocals.

The (formal) Laurent series for the Riemann mapping

$$\mathit{LC}(z) = \frac{1}{z} + c_1+c_2z+c_3z^2+\cdots$$

$$= \frac{1}{z} + m_1+(m_2-m_1^2)z +(m_3 -3m_2m_1+2m_1^3)z^2 + \dotsb$$

has the CI

$$\begin{align*} \mathit{LC}^{(-1)}(z)={} & \frac{1}{z} + \; \frac{m_1}{z^2} + \; \frac{m_2}{z^3} +\;\frac{m_3}{z^4}+\dotsb \\ ={} & \frac{1}{z} + \; \frac{c_1}{z^2} + \; \frac{c_2 + c_1^2}{z^3} +\;\frac{c_3 + 3\; c_2c_1 + c_1^3}{z^4}+\dotsb. \end{align*}.$$

$LC(z)$ has the form of the elliptic modular invariant of eqn. 1.1 of the pdf by Harvey et al. and the q-expansion of the modular invariant j-function of eqn. 10 on p. 6 of "Modular Matrix Models" by Yang-Hui He and Vishnu Jejjala.

The multivariate polynomials in the numerators of $\mathit{LC}^{(-1)}(z)$ are the partition polynomials of A134264, which I usually call the noncrosssing partition polynomials or the refined Narayana partition polynomials or the refined h-vectors of the associahedra but also the free cumulant partition polynomials or Voiculescu polynomials of free probability / random matrix theory. See, e.g., eqns. 35 an 36 on p. 11 of H&J. (See also "Three lectures on free probability" by Novak and LaCroix and "A Simple Introduction to Free Probability Theory and Its Application to Random Matrices" by Xia.)

Of the various sets of CI partition polynomials I've come across, this is the only one that is an Appell Sheffer polynomial sequence in a distinguished indeterminate. Taking the derivative w.r.t. $c_1$ gives a simple spot check of this assertion, illustrating the iconic property of Appell sequences, the lowering operation

$$D_{c_1} P_n(c_1,c_2,...,c_n) = n \cdot P_{n-1}(c_1,c_2,...,c_{n-1}),$$

e.g.

$D_{c_1}P_3(c_1,c_2,c_3) = D_{c_1}(c_3 + 3\; c_2c_1 + c_1^3) = 3(c_2+c_1^2) = 3P_2(c_1,c_2)$.

So, $D_{c_1}$ is a lowering operator. The umbral calculus formalism of Appell sequences then implies

$$(P.(c_1,c_2,...) + \alpha)^n = \sum_{k=0}^n \binom{n}{k}P_k(c_1,..,c_k) \alpha^{n-k}= P_n(c_1+\alpha, c_2,...)$$

as long as $\alpha$ commutes with all the indeterminates in multiplication, analogous to the Hermite and Bernoulli polynomials. An Appell raising op can be formed as well and this gives rise to an associated differential evolution equation for the e.g.f. of the sequence--in umbral notation, $e^{tP.(c_1,c_2,...)} = e^{tP.(0,c_2,...)}e^{tc_1}$--and recursion relations.

The numerator polynomials of $\mathit{LC}(z)$ are the free moment partition polynomials of A350499. They do not form an Appell sequence in a distinguished indeterminate.

On a more basic level, the SL_2 group is related to the ops $D_z, zD_z,$ and $z^2D_z$, which can be related via normal ordering to the Pascal, Bell, and factorial-normalized Laguerre polynomials of order -1, or the Lah polynomials, all binomial Sheffer sequences with associated binomial convolution properties and raising and lowering ops easily described with umbral calculus in terms of umbral compositional inverses (or pairs of invertible triangular matrices). See, e.g., this MO-A, this MSE_A I, and this MSE-A II. The associated Laguerre polynomial diff ops, which can be derived from the Laguerre diff ops of order -1 by conjugation, are related to automorphic/modular forms and to general Sheffer polynomial sequences and, therefore, to umbral calculus as well.

Naturally, these machinations are also related to the Dedekind eta function. See my pdf in my post "Infinigens, the Pascal Triangle, and the Witt and Virasoro Algebras" and search on Dedekind.

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