[This question](https://math.stackexchange.com/questions/3758097/are-any-interesting-classes-of-polynomial-sequences-besides-sheffer-sequences-gr) on math.stackexchange.com has 35 views, three up-votes, and not a word from anybody, so I'm posting it here.


Let us understand the term **polynomial sequence** to mean a sequence $(p\_n(x))\_{n=0}^\infty$ in which the degree of $p\_n(x)$ is $n.$


The **umbral composition** $((p\_n\circ q)(x))\_{n=0}^\infty$ (not $((p\_n\circ q\_n)(x))\_{n=0}^\infty$) of two polynomial sequences $(p\_n(x))\_{n=0}^\infty$ and $(q\_n(x))\_{n=0}^\infty,$ where for every $n$ we have $p\_n(x) = \sum\_{k=0}^n p\_{nk} x^k,$ is given by
$$
(p\_n\circ q)(x) = \sum\_{k=0}^n p\_{nk} q\_k(x).
$$


An **Appell sequence** is a polynomial sequence $(p\_n(x))\_{n=0}^\infty$ for which $p\,'\_n(x) = np\_{n-1}(x)$ for $n\ge1.$


A **sequence of binomial type** is a polynomial sequence $(p\_n(x))\_{n=0}^\infty$ for which $$ p\_n(x+y) = \sum\_{k=0}^n \binom n k p\_k(x) p\_{n-k}(y) $$
for $n\ge0.$


A **Sheffer sequence** is a polynomial sequence $(p\_n(x))\_{n=0}^\infty$ for which the linear operator from polynomials to polynomials that is characterized by $p\_n(x) \mapsto np\_{n-1}(x)$ is shift-equivariant. A shift is a mapping from polynomials to polynomials that has the form $p(x) \mapsto p(x+c),$ where every term gets expanded via the binomial theorem.


At least since around 1970, it has been known that


* Every Appell sequence and every sequence of binomial type is a Sheffer sequence.
* The set of Sheffer sequences is a group under umbral composition.
* The set of Appell sequences is an abelian group under umbral composition.
* The set of sequences of binomial type is a non-abelian group under umbral composition.
* The group of Sheffer sequences is a semi-direct product of those other two groups.
* For every sequence $a\_0, a\_1, a\_2, \ldots$ of scalars there is a unique Appel sequence $(p\_n(x))\_{n=0}^\infty$ for which $p\_n(0) = a\_n$ for $n\ge0.$
* For every sequence $c\_1, c\_2, c\_3, \ldots$ of scalars there is a unique sequence $(p\_n(x))\_{n=0}^\infty$ of binomial type for which $p\,'\_n(0) = c\_n$ for $n\ge1.$ This can be proved by induction on $n.$ (And in every case $p\_0(0)=1$ and $p\_n(0)=0$ for $n\ge 1.$)


**So my question is** whether Sheffer sequences exhaust the list of interesting classes of polynomial sequences that are groups under this operation? Are there any others of interest?



Are any interesting classes of polynomial sequences besides Sheffer sequences groups under umbral composition?
Another equivalent characterization of Sheffer sequences is that they fit into a generating function of the form
$$\sum\_{n=0}^{\infty}\frac{p\_n(x)}{n!}t^n=f(t)e^{xg(t)}.$$
Most of the results on Sheffer sequences apply to a more general setting where we work with a function $\Psi(x)=\sum\_{n\geq 0}x^n/c\_n$ and define $\Psi$-Sheffer sequences as those which satisfy a generating function of the form
$$\sum\_{n=0}^{\infty}\frac{p\_n(x)}{c\_n}t^n=f(t)\Psi(xg(t)).$$
These $\Psi$-Sheffer sequences also form a group under Umbral composition and this group is also a semidirect product of its $\Psi$-Appell subgroup and $\Psi$-binomial type subgroup. It should be noted that abstractly these groups are all isomorphic no matter the choice of $\Psi$: Let $A$ be the group of invertible power series $\mathbb C[[x]]^{\times}$ under multiplication and let $B$ be the (nonabelian) group $x\mathbb C[[x]]^{\times}$ under composition. We can let $B$ act on $A$ by composition and the result is that the group of $\Psi$-Sheffer sequences is isomorphic to the semidirect product of $B$ and $A$.


The details and proofs can be found in Steven Roman's papers "The Theory of the Umbral Calculus I-III", where he gives lots of examples of families of special polynomials that can be treated by this new umbral setting: Chebyshev, Jacobi, Gegenbauer etc. For a treatment that is a more modern you can see S. Zemel ["Generalized Riordan groups and operators on polynomials"](https://www.sciencedirect.com/science/article/pii/S0024379516000240#se0060).


