Sheffer polynomials $\{P\_n(x)\}$ have generating function $P(x,t) = \sum\_{n=0}^{\infty}P\_n(x)t^n=A(t)e^{xu(t)}$.


This form reminds me of the Lie group–Lie algebra correspondence. Is there any connection to Lie theory, obvious or not?


(I've deleted my reference to orthogonal polynomials, which was incorrect and irrelevant here.)


Here are some articles on Sheffer polynomials:


* [Daniel Galiffa, Tanya Riston. An elementary approach to characterizing Sheffer A-type 0 orthogonal polynomial sequences. 2015.](https://doi.org/10.2140/involve.2015.8.39)
* [Dongsu Kim, Jiang Zeng. A Combinatorial Formula for the Linearization Coefficients of General Sheffer Polynomials. 2001.](https://doi.org/10.1006/eujc.2000.0459)



How are Sheffer polynomials related to Lie theory?
First, a general set of Sheffer polynomials is not orthogonal with respect to some weight function; for example, the prototypical sequence $p\_n(x) = x^n$, which belongs to both special sub-groups of Sheffer polynomials—the Appell and binomial Sheffer polynomials—with the e.g.f. $e^{xt}$, is not an orthogonal set—neither is the celebrated Bernoulli Appell Sheffer sequence. (Usually the Sheffer polynomials are defined by $A(t) e^{xB(t)} = \sum\_{n\geq 0} p\_n(x) \frac{t^n}{n!}= e^{tp\_\cdot(x)}$. The normalization $P\_n(x)=n!p\_n(x)$ often serves to give integer coefficients.)


However, there are connections to constructs associated to Lie theory:


***1)*** The binomial Sheffer sequences have e.g.f.s of the form $e^{xh(t)}$, where $h(0)=0$ and $h'(0) \neq 0$. A dual sequence, its umbral compositional inverse, has the e.g.f. $e^{x\;h^{(-1)}(t)}$ defined by the compositional inverse function. As formulated by Charles Graves as early as 1853, with $g(z) = \frac{1}{\partial\_z h(z)}$, the action on a function $f(z)$ analytic at $z$ of the exponential map of the infinitesimal generator (infinigen)


$$ g(z)\frac{\partial }{\partial z} = \frac{1}{h'(z)}\frac{\partial }{\partial z} = \frac{\partial }{\partial h(z)} = \frac{\partial }{\partial \omega} $$


is


**(Eqn. 1)**


\begin{align}
& e^{t\;g(z)\frac{\partial }{\partial z}}\; f(z) = e^{t \;\frac{\partial }{\partial \omega}}\; f(h^{(-1)}(\omega)) \\
= {} & f[h^{(-1)}(\omega + t)] =f[h^{(-1)}(h(z)+t)]. \tag{Eqn. 1}
\end{align}


Then


$$e^{t\;g(z)\partial\_z}\;z \Big\rvert\_{z=0}= h^{(-1)}(t),$$


so the e.g.f. for the associated binomial Sheffer sequence is


$$\left. e^{t\;p.(x)}= e^{x\;h^{(-1)}(t)}= e^{t\;g(z)\partial\_z} e^{zx} \;\right\rvert\_{z=0}.$$


Edit Oct. 16, 2022: (Start)


Then


$$p\_n(x) = (g(z)\partial\_z)^n\; e^{zx} \;\Big\rvert\_{z=0}$$


$$= e^{-x} (g(y-1)\partial\_y)^n e^{yx} \; |\_{y=1}= \left. e^{-x} \left[x \;g\left(\frac{u}{x}-1\right)\partial\_u\right]^n e^{u} \; \right|\_{u=x}.$$


Specializing to $g(z)=(1+z)^{m+1}$, allows a general connection between the normal ordering of iterates of the Scherk-Witt-Lie vectors / infinigens $z^{m+1}\partial\_z$ and well-known binomial Sheffer sequences. Then


\begin{align}
p\_n^{(m)}(x) & =((1+z)^{m+1}\partial\_z)^n \;e^{zx}\;\Big|\_{z=0} \\[6pt]
& = e^{-x}\; (y^{m+1}\partial\_y)^n \;e^{yx} \; \Big|\_{y=1} \\[6pt]
& = e^{-x}\left. \left(x \left(\frac{u}{x} \right)^{m+1} \partial\_u\right)^n\; e^u \right|\_{u=x} \\[6pt]
& = e^{-x}\; x^{-mn} \;(u^{m+1}\partial\_u)^n\; e^{u} \; \Big|\_{u=x} \\[6pt]
& = e^{-x}\; x^{-mn} \;(x^{m+1}\partial\_x)\;^n e^x = e^{-x}\; p\_n^{(m)}(:x\partial\_x:)\; e^x
\end{align}


with, by definition, $(:x\partial\_x:)^n := x^n\partial\_x^n$, implying


**(Eqn. 2a)**


\begin{align}
(x^{m+1} \partial\_x)^n & = x^{mn} p\_n^{(m)}(:x\partial\_x:) \\[6pt]
& = x^{mn}\; m^n St1\_n^r \left( \frac{x\partial\_x}{m} \right) \tag{Eqn. 2a}
\end{align}


where the e.g.f. for the $m$-th family of binomial Sheffer polynomials $p\_n^{(m)}(x)$ is


**(Eqn. 2b)**


\begin{align}
& e^{tp.^{(m)}(x)} = e^{h^{(-1)}(t)x} \\[6pt]
= {} & \exp[((1-mt)^{-\frac{1}{m}}-1)x] \tag{Eqn. 2b}
\end{align}


and $St1\_n^{r}(x)$ are the reversed unsigned Stirling polynomials of the first kind of OEIS [A094638](https://oeis.org/A094638) (see also [A008275](https://oeis.org/A008275), [A048994](https://oeis.org/A048994), and [A130534](https://oeis.org/A130534)).


For the special linear Lie algebra $sl\_2$:


For $m =-1$, the infingen $g(z)\partial\_z = g(1+z)\partial\_z= \partial\_z$ has associated $h(z)=h^{(-1)}(z)=z$ giving the **translation** $e^{t\partial\_z}\;f(z) = f(z+t)$, consistent with Eqn. (2) with $p^{(-1)}\_n(z) = z^n$, the iconic Appell Sheffer sequence, which is also a binomial Sheffer sequence.


For $m=0$, the infingen $g(z)\partial\_z = z\partial\_z$ has associated $h(z)=\ln(z)$ and $h^{(-1)}(\omega)=e^\omega$ giving, from Eqn. (1), the **scaling** $e^{tz\partial\_z}f(z) = f[\exp(\ln(z)+t)] = f(e^tz)$. Eqn. (2) associated with $g(z)=(1+z)$, $h(z) =\ln(1+z)$, and $h^{(-1)}(\omega) =e^{\omega}-1$ gives


$$(z\partial\_z)^n = p^{(0)}\_n(:z\partial\_z:) = \operatorname{St2}\_n(:z\partial\_z:), $$


the Bell / Touchard / Scherk / Stirling polynomials of the second kind with the e.g.f. $\exp[(e^t-1)x]$ (cf. [A008277](https://oeis.org/A008277) and [A048993](https://oeis.org/A048993)). This is easy to corroborate using $(z\partial\_z)^n z^k = k^nz^k$.


For $m=1$, the infingen $g(z)\partial\_z = z^2\partial\_z$ has associated $h(x)=-\frac{1}{z}$ and $h^{(-1)}(\omega)=-\frac{1}{\omega}$ giving, from Eqn. (1), the **vertical shearing** $e^{tz^2\partial\_z}f(z) = f\left [-\frac{1}{-\frac{1}{z}+t}\right] = f(\frac{z}{1-tz})$. Eqn. (2) associated with $g(z)=(1+z)^2$, $h(z) =\frac{z}{1+z}$, and $h^{(-1)}(\omega) =\frac{\omega}{1-\omega}$ gives


$(z^2\partial\_z)^n =z^n p^{(1)}\_n(:z\partial\_z:) = z^n Lah\_n(:z\partial\_z:)$,


the shifted Lah polynomials, or shifted, normalized, unsigned Laguerre polynomials of order -1, $n! \operatorname{Lag}\_n^{(-1)}(-z)$, with the e.g.f. $\exp[z \frac{t}{1-t}]$ (cf. [A008297](https://oeis.org/A008297) and [A111596](https://oeis.org/A111596)). This is corroborated by noting


$$(z^2\partial\_z)^n = z\;(z\partial\_zz)^n\;z^{-1} = z\;z^n\partial\_z^n z^n \;z^{-1} = z^n\; z \;(\partial\_z^nz^n)\; z^{-1} = z^n\; n!\; z\; \operatorname{Lag}\_n(:z\partial\_z:)\;z^{-1}$$


from the Rodriguez formula for the associated Laguerre polynomial sequences.


(End)


***2)*** Each Sheffer sequence has a pair of ladder ops—the raising/creation and lowering/destruction/annihilation ops defined by $L \; p\_n(x) = n \; p\_{n-1}(x)$ and $R \; p\_n(x) = p\_{n+1}(x)$—satisfying the Graves–Lie bracket of vector fields, the commutator, relation


$$[L,R] = 1,$$


from which the Graves–Pincherle derivative


$$[f(L),R] = f'(L)$$


follows.


There is also the commutator


$$[(g(z)\partial\_z)^n,u(z)] = n \; (g(z)\partial\_z)^{n-1}$$


for the powers of the Lie derivative / infinitesimal generator for the Scherk-Comtet partition polynomials of [A139605](https://oeis.org/A139605), forming an underlying calculus for the binomial Sheffer sequences.


Edit July 1, 2023: (Start)


The typo $u(z)$ just above should be $h(z)$ so that


$$[(g(z)\partial\_z)^n,h(z)] = n \; (g(z)\partial\_z)^{n-1}.$$


Proof:


$$e^{tg(z)\partial\_z} \;h(z)f(z) - h(z) \;e^{tg(z)\partial\_z}\;f(z)$$


$$= (h(z)+t)\; f(h^{(-1)}(h(z)+t)) -h(z)\; f(h^{(-1)}(h(z)+t))$$


$$= t\; f(h^{(-1)}(h(z)+t)) = t e^{tg(z)\partial\_z} \;f(z)$$


$$=\sum\_{n \geq 0} (g(z)\partial\_z)^{n} \frac{t^{n+1}}{n!} \;f(z) =\sum\_{n \geq 1}n (g(z)\partial\_z)^{n-1} \frac{t^n}{n!}\;f(z) $$


(End)


***3)*** A general Sheffer sequence (a semidirect product of the Appell and binomial Sheffer sequences) is associated with a generalized Lie derivative via


$$e^{t(q(z)+g(z)\partial\_z)} \; e^{xz} |\_{z=0} = A(t) e^{xf^{(-1)}(t)}$$


with $q(z) = \partial\_t \ln(A(t)) \;|\_{t=f(z)}$ and $g(z) = 1/f'(z)$.


I have an extensive set of posts at my web blog and links to numerous OEIS entries and MO-Q&As and MSE-Q&As related to this topic. All the symmetric polynomials/functions--complete, elementary, power, Faber--are related to Sheffer Appell sequences with their related ladder ops. The Faa di Bruno/Bell and cycle index polynomials of the symmetric groups, a.k.a the refined Stirling partition polynomials of the second and first kinds, and other compositional partition polynomials are all Appell Sheffer sequences in a distinguished indeterminate. The closely related sets of Lagrange inversion partition polynomials, including the refined Euler characteristic partition polynomials of the associahedra, all have the raising op $g(z)\partial\_z$, as shown above--one set, OEIS [A134264](https://oeis.org/A134264), is an Appell sequence and is related to free probability, inversion of Laurent series, and characterization of Kac-Schwarz operators related to Heisenberg-Virasoro groups. Multiplicative inversion is intimately bound with the refined Euler characteristic partition polynomials of the permutahedra and the operational (differential/matrix) calculus of Appell Sheffer polynomials. The Bernoulli Appell polynomials are of course related to the BCHD theorem, exponential mappings of the Lie commutator, topology, Todd operator and class of Hirzebruch, the Hurwitz zeta function, and more. Formal group laws (and generalized local Lie groups) are related to the linearization coefficients of products of binomial Sheffer polynomials. The list goes on.


The orthogonal Sheffer sequences such as the associated Laguerre polynomials have a lot of underlying group theory and vector fields associated with them. Vilenkin with "[Special Functions and the Theory of Group Representations](https://doi.org/10.1090/mmono/022)", Talman, Miller, Gilmore, Feinsilver, and many others have written books on the underlying group theory. (None covers every aspect.)


One group of researchers has also written extensively on this topic. See, e.g., "[One-parameter groups and combinatorial physics](https://arxiv.org/abs/quant-ph/0401126)" by
Duchamp, Penson, Solomon, Horzela, and Blasiak (although quite negligent of reffing the OEIS--tribal instincts...sigh...What can you do?) as well as Wolfdieter Lang.


**4)** (Added Oct. 16, 2022) The raising operator of an Appell Sheffer sequence gives the infinitesimal generator, related to the Riemann zeta function, for Heaviside fractional differ-integral operators. See, e.g., my MO-Q "[Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus](https://mathoverflow.net/questions/111165/riemann-zeta-function-at-positive-integers-and-an-appell-sequence-of-polynomials)" and MO-A to "[What's the matrix of logarithm of derivative operator (lnD)? What is the role of this operator in various math fields?](https://mathoverflow.net/questions/382735/whats-the-matrix-of-logarithm-of-derivative-operator-ln-d-what-is-the-rol/383563#383563)".


