Are any interesting classes of polynomial sequences besides Sheffer sequences groups under umbral composition? This question 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?
 A: 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".
