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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?

Michael Hardy
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    Among well known Appell sequences are the Hermite polynomials and the Bernoulli polynomials. Among well known sequences of binomial type are the rising factorials and the falling factorials. $\qquad$ – Michael Hardy Jul 15 '20 at 18:45

1 Answers1

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If you consider the family of polynomial sequences $((p_{a, n})_{n=0}^\infty)_{a>0}$ such that $p_{a, n}(x) = x^n + a$, then

$$ (p_{a,n} \circ p_b)_n(x) = (x^n + b) + a(x^0+b) = p_{a+b+ab,n}(x) $$

so the family is closed under umbral composition, however they are not Sheffer sequences as their generating function is

$$ \sum_{n=0}^\infty \frac{p_{a,n}(x)}{n!} t^n = e^{xt} + ae^t $$ which is not of the form $A(t)e^{xB(t)}$. This example demonstrate that there are probably many classes of polynomials stable under umbral composition without being Sheffer sequences, but they are probably not as "interesting" and here is why :

We study sheffer sequences because the map $Q$ such that $Qp_n(x) = np_{n-1}(x)$ is shift-invariant, as you mentioned. When $Q$ is shift-invariant, $Q$ is called a delta operator. A shift invariant operator can always be written as a power series in the derivative operator. This power series is called the indicator. Furthermore, umbral composition of sheffer sequences have as delta operator the composition of the indicators evaluated in the derivative operator. But for non-sheffer sequences, we can't define an indicator for $Q$ because it is not shift-invariant. Therefore umbral composition makes less sense. So there are probably no other classes of polynomial sequences closed under umbral composition that are "interesting".

Nolord
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