In this paper, we explore the effectiveness of almost purely operational methods in the study of umbral calculus. To accomplish this goal, we systematically reconstruct the theory operationally, offering new proofs and results throughout. Our approach is applied to the study of invertible power series, where we notably offer a concise two-line proof of Lagrange's inversion theorem and derive formulas for both fractional and regular compositional iterates, for which the latter is new. In addition, we will develop several new insights into umbral operators, including their fractional exponents and novel expressions, in connection with the previously discussed applications. Finally, we will elaborate on both established and new concepts and examples, such as pseudoinverses of delta operators and a natural generalization of the Laguerre polynomials.

§ INTRODUCTION

At its inception, umbral calculus comprised a set of informal but powerful techniques devised by Blissard in the late 19th century. These techniques mainly involved manipulating the indices and exponents of certain number sequences, particularly Bernoulli numbers, while exploiting the properties of exponents. Despite yielding new identities, these techniques initially faced skepticism from the scientific community due to their lack of rigor.

In the early 1970s, Rota et al. [25, 30] laid the first rigorous foundation for umbral calculus through its integration with Pincherle and Amaldi's ideas on linear operator [28] and the theory of Sheffer sequences. The latter was a theory about certain polynomial sequences, independently developed in different directions by Sheffer [33] and Steffensen [35], generalizing that of Appell [2]. Consequently, numerous novel frameworks for the study of umbral calculus emerged. Notably, in collaboration with Roman, Rota further developed the notion in [29] that the informal methods of the nineteenth century could be elucidated by linear forms over the set of polynomials, referred to as linear functionals. As the theory evolved, umbral calculus increasingly diverged from its origins, prompting Rota and Taylor to construct what they designated as the Classical umbral calculus [31], representing the first successful axiomatic foundation for the umbral calculus of the 19th century.

The aim of this manuscript is to extend the foundational work established by Rota in his seminal papers with a closer study of the coefficients of basic polynomial sequences and an emphasis on the use of linear operators. This new methodology, coupled with new convenient notations, will enable the derivation of unexpectedly new results.

In <Ref>, we will follow Rota's initial approach to develop the fundamental tools from [30], starting with the definition of delta operators, while incorporating a more systematic use of linear operators. For example, we will introduce bivariate characterizations of the shift-invariant, umbral, and Sheffer operators, which will simplify some of the proofs. In <Ref>, we will conduct an in-depth study of the coefficients of basic sets and, based on Labelle's insights [20], apply them to the iterative theory [38] to uncover new proofs of Jabotinsky's formulas for the iterative logarithm and fractional iterates of the power series $f(x) = \sum_{n\geq1} a_n x^n/n!$ when $a_1 = 1$. Additionally, we will swiftly prove a potentially new formula for regular iterates when $a_1$ is any other nonzero constant

\[
\frac{f^s(x)^k}{k!} = \sum_{n=1}^\infty \frac{x^n}{n!} \sum_{k=j_0 \leq \ldots \leq j_s = n} \prod_{i=0}^{s-1} B_{j_{i+1}, j_i}(a_1, \ldots, a_{j_{i+1}-j_i+1}).
\]

where $B$ is the partial Bell polynomials, $k$ and $s$ are natural integers.

In <Ref>, these formulas, together with operational identities, will be later applied to derive closed forms for umbral operators and initiate a discussion on their fractional exponents. Following that, we identify a particular class of umbral operators that fulfill a natural operator identity. We will then construct pseudoinverses for delta operators, we coined sigma operators, which have application in a surprising new definition for the generalization of summation with fractional bounds. Finally, numerous classic and new examples will be discussed in <Ref>.

§ DEFINITIONS AND FIRST RESULTS

This section introduces the main objects of study: delta,  umbral and Sheffer operators. Their construction will be followed by proofs of their basic properties, which will enhance our understanding of them. These insights will pave the way to demonstrate the equivalence between basicness and binomial type for polynomial sequences.

§.§ Preliminaries

Let $\K$ (typically $\R$ or $\C$) be a field of characteristic zero. In this paper, our focus lies on the $\K$-algebra of univariate polynomials, denoted by $\P$, or on the $\K$-algebra of formal univariate power series, denoted by $\tP$. Within $\P$, we define our linear operators. The definitions for the initial operators are provided in Figure <ref>.

Evaluation at zerof(0)
Evaluation $(a \in \K)$af(a)
Scalar ($c \in \K$)ccf(x)
Multiplicationx f(x)

The elementary operators

It is customary to represent the composition and the application linear function as an implicit product, such that $(\x \circ D)(f)(x)$ is more succinctly expressed as $\x D f(x)$. This notation is particularly intuitive when viewing linear operators as a generalization of scalar multiplication. By slight abuse of notation, we denote the identity as $1$ rather than $I$, as it can be interpreted as a special case of the scalar operator with $c = 1$.

Readers should note that, depending on the context, we will utilize the evaluation operator $\Ev{a}$ either as an endomorphism (mapping a function to a constant function) or as a linear form (a linear operator mapping to the field $\K$). Additionally, in multivariate settings, we will distinguish linear operators by indexing the variable to which they apply. By default, operators are applied to the variable $x$. In other words, if $U$ is a linear operator, $U_x = U$. Moreover, for $n \in \N$, we define the $n$-th coefficient in the Taylor series of $f(x)$ using the common notation $[x^n]$. In our linear operators, this can be expressed as

\begin{equation}
[x^n] f(x) = \inv{n!} \E D^n f(x).
\end{equation}

For a formal power series $f(x) = \sum a_n x^n$, we define its order and degree as the index of the first and last nonzero coefficients, respectively. We denote them by “$\ord$” and “$\deg$”, and adhere to the standard convention $\deg 0 = -\infty = - \ord 0$.

Although we will not discuss the topology and convergence of formal power series, we should mention that it can be successfully described by the ultrametric space induced by the norm $\norm f = 2^{-\ord(f)}$.

For $p \in \N^*$, where $\N^*$ denotes the set of natural numbers excluding zero, we differentiate the notation for iteration and exponentiation,

\[
f^p(x) = f(f^{p-1}(x)) \quad\et\quad f(x)^p = f(x) \cdot f(x)^{p-1},
\]

with $f^0(x) = x$ and $f(x)^0 = 1$.
We will employ the generalized exponentiation operation for linear operators, introduced by the author in [4] and defined as

\begin{equation}\label{e:U^V}
U^V = \sum_{n=0}^\infty (U-1)^n \binom{V}{n},
\end{equation}

where $\binom{x}{n} = \frac{(x)_n}{n!}$ represents the generalized binomial coefficient, and $(x)_n = x(x-1) \cdots (x-n+1)$ denotes the falling factorial. It's important to note that in general

\begin{equation}
U^V \neq \sum_{n=0}^\infty \binom{V}{n} (U-1)^n,
\end{equation}

as linear operators do not always commute.

§.§ Polynomial sequences and invertible operators

For $n\in\N$, let $\P_n$ be the set of polynomials with a single variable $x$ of degree less than $n$. Define $\hP_n \defeq \P_n \setminus \P_{n-1}$ to represent the subset of polynomials of degree exactly $n$ (with the understanding that $\P_{-1} = \varnothing$). We denote by $\L(S)$ the set of endomorphisms over a set $S$.

A sequence of polynomials $(p_n)_{n\in\N}$ is said to be a polynomial sequence $\{p_n\}_{n\in\N}$ if, for all positive integers $n$, $p_n(x) \in \hP_n$. It is self-evident that $(p_k)_{0\leq k\leq n}$ is a basis for $\P_n$, leading to the following proposition.

If $U, V \in \L(\P)$ then $U = V$ if and only if for some polynomial sequence $\{p_n\}_{n\in\N}$, $U p_n(x) = V p_n(x)$ for all positive integers $n$.

Now we define

\begin{equation}
\hL \defeq \bigcap_{n\in\N} \L(\hP_n) \subset \L(\P)
\end{equation}

the set of linear operators that preserve the polynomial degree upon application.

A polynomial sequence $\{p_n\}_{n\in\N}$ is said to be unitary when $[x^n] p_n(x) = 1$ for all $n\in\N$.

For every operator in $\phi \in \hL$, we define its associated polynomial sequence $\{ \phi_n \}_{n\in\N}$ by $\phi_n(x) \defeq \phi x^n$. As $\phi$ preserves the degree of polynomials, it induces a transformation between bases of $\P$. Consequently, the ensuing proposition is valid.

If $\phi \in \hL$, then $\phi$ has an inverse $\phi^{-1}$.

This proposition indicates the presence of a division ring structure in $\hL$. Furthermore, the choice of mapping monomials by $\phi$ to their corresponding polynomial sequences entails no loss of generality. Indeed, selecting an alternative sequence, such as $\{p_n\}_{n\in\N}$, would yield an identical analysis, but with umbral operators of the form $\phi p^{-1}$.

A special type of polynomial sequence that can be subjected to study via umbral calculus is polynomial sequences of binomial type.

We say $\{p_n\}_{n\in\N}$ is of binomial type if for all $x, y$ and $n\in\N$

\begin{equation}\label{e:binom}
p_n(x+y) = \sum_{k=0}^n \binom{n}{k} p_k(x) p_{n-k}(y).
\end{equation}

It is worth noting that there are two ways to view this functional equation. By setting $q_n(x) \defeq p_n(x)/n!$, we obtain a convolution identity

\begin{equation}\label{e:convo}
q_n(x+y) = \sum_{k=0}^n q_k(x) q_{n-k}(y).
\end{equation}

Both conventions are used in the literature [30, 12], but we will demonstrate that the original convention is operationally more convenient.

For $\phi \in \hL$, the application $\phi \mapsto \{\phi_n\}_{n\in\N}$ is bijective.

For the surjectivity, suppose that we have a polynomial sequence $\{p_n\}_{n\in\N}$. We define $\phi$ by $\phi(x^n) = p_n(x)$ extended by linearity to power series. By definition, $\phi$ does not affect the degree, so $\phi \in \hL$.

For injectivity, suppose $\phi, \psi \in \hL$ such that $\phi_n(x) = \psi_n(x)$ for all $n$ and $x$. Equivalently, $\phi x^n = \psi x^n$ for all $n$, and hence $\phi = \psi$ by <Ref>. Thus, the map is injective.

We designate $p \in \hL$ as the inverse of the map in <Ref> for $\{p_n\}_{n\in\N}$, and extend the notion of unitarity to operators in $\hL$ by saying that $p$ is unitary if and only if $\{p_n\}_{n\in\N}$ is unitary.

We aim to present bivariate characterizations of some known classes of operators. It is observed that many of the operators studied in umbral calculus are characterized in bivariate settings.  A notable instance of such an operator is the linear operator that maps monomials to sequences of binomial-type polynomials.

A polynomial sequence $\{p_n\}_{n\in\N}$ is of binomial type if and only if, for two variables $x, y$, $p_{x+y} = p_x p_y$.

On the one hand, $p_{x+y}(x+y)^n = p_n(x+y)$. On the other hand
\[
p_x p_y(x+y)^n = p_x p_y\sum_{k=0}^n \binom{n}{k} x^k y^{n-k} = \sum_{k=0}^n \binom{n}{k} p_k(x) p_{n-k}(y) = p_n(x+y).
\]

§.§ Delta operators

The construction of the original umbral calculus [25, 30] was performed using delta operators. While we believe this is not the most natural way to construct the theory, it is arguably the simplest and the easiest to motivate.

An operator $U \in \L(\P)$ is said to be shift-invariant if it commutes with $E^a$ for all $a$.

The simplest example of shift-invariant operators is, of course, $E^a$. Another example is the derivative operator $D$. However, the multiplication operator $\x$ is not shift-invariant.

The following proposition provides a bivariate characterization of shift-invariant operators.

A linear operator $U$ is shift-invariant if and only if, for two variables $x$ and $y$, $U_x = U_{x+y}$.

The equivalence is explained by the general identity for linear operators: $E_x^y U_x = U_{x+y} E_x^y$.

An operator $Q$ is a delta operator if $Q$ is shift-invariant and $Qx = c$ for some constant $c \neq 0$. If $c = 1$, we say that $Q$ is unitary.

After the derivative operator $D$, the second most important (unitary) delta operator is the forward difference operator $\Delta$, defined by

\[
\Delta f(x) = f(x+1) - f(x).
\]

Writing $\Delta = E - 1$, it can be easily checked that $\Delta$ satisfies the definition conditions.

While the definition of delta operators may appear too weak to support a robust mathematical theory, this perception is mistaken, it underscores why this approach to constructing umbral calculus is, in fact, remarkably elegant.

If $Q$ is a delta operator, then

* for $a \in \K$, $Qa = 0$.

* for a nonconstant polynomial $p(x)$, $\deg Qp(x) = \deg p(x) - 1$.

* We make a single use of all the given properties

\[
Qa + c = Qa + Qx = Q(a+x) = Q E^a x = E^a Qx = E^a c = c,
\]

conequently $Qa = 0$.

* For $n>0$, let $q(x) = Q x^n$. For any $a$

\begin{align*}
q(x+a) &= E^a q(x) = E^a Qx^n = QE^a x^n = Q(x+a)^n \\
&= Q \sum_{k=0}^n \binom{n}{k} a^k x^{n-k} = \sum_{k=0}^n \binom{n}{k} a^k Qx^{n-k}.
\end{align*}

Therefore, by applying $\E$, we recover the polynomial $q(a)$, for any $a$

\[
\E q(x+a) = q(a) = \sum_{k=0}^n \binom{n}{k} a^k \E Qx^{n-k}.
\]

The coefficient in front of $a^n$ is $\E Q x^0 = 0$ by (a). The one in front of $a^{n-1}$ is $n \E Q x = nc \neq 0$. Therefore, $q(x)$ is indeed of degree $n-1$.

If $p(x)$ is a polynomial such that $Qp(x) = 0$ then $p(x)$ is constant.

Let us prove the contrapositive and suppose that $p(x)$ is nonconstant, that is, of degree $\geq 1$. Then, by <Ref>, $\deg Qp(x) \geq 0$. In other words, $Qp(x) \neq 0$.

§.§ Sheffer and umbral operators

An invertible operator $s$ is called a Sheffer operator for $Q$ if

\[
Qs_n(x) = n s_{n-1}(x),
\]

We then refer to $\{s_n\}_{n\in\N}$ as a Sheffer sequence or set for $Q$. If $Q = D$, then $s$ is an Appell operator and $\{s_n\}_{n\in\N}$ an Appell sequence or set.

An invertible operator $\phi$ is said to be an umbral operator or basic for $Q$ if it satisfies

* $\{\phi_n\}_{n\in\N}$ is a Sheffer set for $Q$

* $\phi_0(x) = 1$

* for $n>0$, $\phi_n(0) = 0$.

We denote the relation by $\phi \leadsto Q$. We then call $\{\phi_n\}_{n\in\N}$ the basic sequence or set of $Q$. Finally, $\phi$ is said to be a Sheffer/Appell/umbral operator if it is for some delta operator.

The additional conditions imposed on basic sets naturally constrain Sheffer sets, facilitating a bijection with their delta operator.

* For every Sheffer operator $s$, there exists a unique delta operator $Q$ such that $s$ is a Sheffer operator for $Q$.

* There is a one-to-one correspondence between the set of delta operators and the set of umbral operators.

* If $R$ also serves as a delta operator for $s$, then, in particular, for all $n\in\N$, $Q s_n(x) = R s_n(x)$. Since $\{s_n\}_{n\in\N}$ forms a polynomial sequence, by <Ref>, $R = Q$.

* We construct by induction the basic polynomial sequence $\{\phi_n\}_{n\in\N}$ for $Q$. The initial term is uniquely imposed by its definition. Assume, for $n \in \N^*$, that $\{\phi_k\}_{0\leq k<n}$ is uniquely determined and satisfies the basic set conditions. Given that it constitutes a basis for $\P_{n-1}$, the expression $Q x^n \in \P_{n-1}$ (by <Ref>) can be uniquely decomposed with respect to the basis

\[
Qx^n  = \sum_{k=0}^{n-1} c_k \phi_k(x), \quad c_{n-1} \neq 0.
\]

Should $\phi_n(x)$ meet the criteria of the Sheffer condition, it can be reformulated as follows

\[
Q\bra{x^n - \sum_{k=0}^{n-1} \frac{c_k}{k+1} \phi_{k+1}(x)} = 0,
\]

wherein, by <Ref>, the operand of $Q$ must be constant. It is chosen to be 0, ensuring that $\phi_n(0) = 0$ and thereby uniquely defining $\phi_n(x)$ as

\[
\phi_n(x) = \frac{n}{c_{n-1}}\bra{x^n - \sum_{k=1}^{n-1} \frac{c_{k-1}}{k} \phi_k(x)}.
\]

Consequently, we have constructed the unique basic sequence for $Q$. In particular, it is a Sheffer sequence. Therefore, by (a), $Q$ is established as its unique delta operator, thus confirming the bijection.

The basic set for $D$ is evidently the monomials $\{x^n\}_{n\in\N}$. This generalizes straightforwardly for the delta operator $D/\lambda$, where $\lambda \neq 0$. Its basic set is verified to be $\{(\lambda x)^n\}_{n\in\N}$. Since this transformation constitutes a scaling action, we refer to the umbral operator $\str\lambda \leadsto D/\lambda$ as the stretch operator.

We define $\varphi$ via the relation $\varphi\leadsto\Delta$. It can be verified that $\varphi_n(x) = (x)_n$, the falling factorials. Later, we shall develop tools to derive expressions for basic sets, eliminating the need to guess their expression.

If $\phi \leadsto Q$, then these statements are equivalent

* $Q x = c$.

* $[x^n]\phi_n(x) = c^{-n}$.

* $\phi_1(x) = x/c$.

In particular, we have the equivalence between $Q$ being unitary, $\phi$ being unitary, and $\phi_1(x) = x$.

(c $\imp$ a) Follows from $Qx = Q c\phi_1(x) = c\phi_0(x) = c$.

(b $\imp$ c) Is evident because $\phi_1(0)$ is always $0$.

(a $\imp$ b) This follows from the construction of $\phi_n(x)$ we made in the proof of <Ref> combined with the fact that $[x^{n-1}] Q x^n = cn$ (as shown in the proof of <Ref> (b)).

A primary focus of this paper is to characterize Sheffer and basic sets using operator identities. This is straightforwardly achieved for Sheffer sets through a general commutation identity.

An invertible operator $s$ is a Sheffer operator for $Q$ if and only if $Qs = sD$.

This set of equalities essentially proves the equivalence

\[
\begin{array}{ccc}
Q s_n(x) &=& n s_{n-1}(x) \\
\vertequal & & \vertequal \\[-3pt]
Q sx^n &=& s D x^n.
\end{array}
\]

The fact that the monomials form a basic set for $D$ and are also of binomial type is no coincidence, as we will show that all basic sets are of binomial type. Additionally, we will demonstrate that the converse holds true as well.

$\{p_n\}_{n\in\N}$ is a basic set for some delta operator $Q$ if and only if $\{p_n\}_{n\in\N}$ is a binomial type polynomial sequence.

( $\imp$ ) We proceed by induction on $n$. The binomial property (<Ref>) trivially holds for $n=0$.

Let $n > 0$. $Qp_n(x+y) = Q E^y p_n(x) = E^y n p_{n-1}(x) = n p_{n-1}(x+y)$. Assuming the induction hypothesis for $n-1$, that is, $p_{n-1}(x+y) = \sum_{k=0}^{n-1} \binom{n-1}{k} p_k(x) p_{n-1-k}(y)$, where $x$ is the variable and $y$ serves as a parameter. Therefore, we have

\begin{align*}
Q p_n(x+y) &= n \sum_{k=0}^{n-1} \binom{n-1}{k} p_k(x) p_{n-1-k}(y) = \sum_{j=1}^n n\binom{n-1}{j-1} p_{j-1}(x) p_{n-j}(y) \\
&= \sum_{j=1}^n j\binom{n}{j} p_{j-1}(x) p_{n-j}(y) = \sum_{k=1}^n \binom{n}{k} Qp_k(x) p_{n-k}(y).
\end{align*}

Therefore, by linearity

\[
Q\bra{p_n(x+y) - \sum_{k=1}^n \binom{n}{k} p_k(x) p_{n-k}(y)} = 0
\]

In accordance with <Ref>, the argument of $Q$ must be constant. However, upon evaluation at $x=0$, this constant is identified as $p_n(y)$ since $p_k(0) = 0$ for $k > 0$. Additionally, $p_0(x) = 1$. Consequently

\[
p_n(x+y) = \sum_{k=0}^n \binom{n}{k} p_k(x) p_{n-k}(y),
\]

thus finishing the induction.

( $\isimp$ ) Analogous to the proof of <Ref>, we define $Q$ by $Q(p_n(x)) = np_{n-1}(x)$ and $Q(p_0(x)) = 0$ extended by linearity to all power series. In particular, because $p_1(x) = cx$ for some nonzero constant $c$, $Qx = Q p_1(x)/c = p_0(x)/c = 1/c \neq 0$. For the shift-invariance, we invoke the binomial property

\begin{align*}
Q E^y p_n(x) &= Q p_n(x+y) = \sum_{k=0}^n \binom{n}{k} Q p_k(x) p_{n-k}(y) = \sum_{k=0}^n \binom{n}{k} k p_{k-1}(x) p_{n-k}(y) \\
&= \sum_{j=0}^{n-1} n \binom{n-1}{j} p_j(x) p_{n-j-1}(y) = n p_{n-1}(x+y) = E^y Q p_n(x),
\end{align*}

and by linearity, $QE^y = E^y Q$, confirming that $Q$ is a delta operator for $\{p_n\}_{n\in\N}$.

[Chu-Vandermonde identity]
The binomial property for $\{(x)_n\}_{n\in\N}$ gives

\[
(x+y)_n = \sum_{k=0}^n \binom{n}{k} (x)_k (y)_{n-k},
\]

which, in the fashion of <ref>, can be rewritten into Chu's generalization of the Vandermonde identity

\[
\binom{x+y}{n} = \sum_{k=0}^n \binom{x}{k} \binom{y}{n-k}.
\]

§ CLASSIC THEOREMS

With the foundational layer of umbral calculus established, we shall now present the first set of computational tools, referred to as the “classic theorems”. These tools, which were central to the groundbreaking work of Rota [30], include the expansion and isomorphism theorems, the Pincherle derivative and closed forms.

§.§ Shift-invariant operator expansion

Undoubtedly, one of the most important early results of umbral calculus is the Expansion Theorem. It expresses every shift-invariant operator as a power series in a delta operator. Such formulas are free from convergence issues in $\P$, as their application to polynomials invariably yields a finite sum, ranging from 0 to the polynomial's degree. Remarkably, this property also applies to any power series that would typically be divergent.

$Q$ is a delta operator with basic set $\{p_n\}_{n\in\N}$ if and only if for all shift-invariant operators $U$

\[
U = \sum_{k=0}^\infty a_k \frac{Q^k}{k!}, \quad a_k = \E U p_k(x).
\]

( $\imp$ ) We apply $\E U$ to the binomial identity

\[
\E U p_n(x+y) = \sum_{k=0}^n \binom{n}{k} \E U p_k(x) p_{n-k}(y) = \sum_{k=0}^\infty \inv{k!} \E U p_k(x) Q_y^k p_n(y),
\]

where the summation index is extended extended to infinity, as the additional terms are zero. By <Ref> and the inherent commutativity of linear operators applied to different variables, we establish

\[
\E_x U_x p_n(x+y) = \E_x U_y p_n(y+x) = U_y \E_x p_n(y+x) = U_y p_n(y).
\]

Setting $a_k = \E U p_k(x)$, we obtain the desired result by linearity.

( $\isimp$ ) Conversely, we initially show that $Q$ is a delta operator. Using the given hypothesis with $E^a$ for any $a$, gives

\[
E^a = \sum_{k=0}^\infty p_k(a) \frac{Q^k}{k!}. \tag{$*$}
\]

This demonstrates the shift-invariance of $Q$. Apply $(*)$ to $x$

\[
x+a = \sum_{k=0}^\infty p_k(a) \frac{Q^k}{k!} x.
\]

Since $\{p_n\}_{n\in\N}$ forms a polynomial sequence and thus a basis, the identification of the coefficient before $p_1(a)$ leads to the conclusion that $Qx$ must be non-zero. Therefore, $Q$ is a delta operator associated with a unique basic set $\{q_n\}_{n\in\N}$. Applying $(*)$ to $q_n(x)$ yields

\[
q_n(x+a) = \sum_{k=0}^n \binom{n}{k} p_k(a) q_n(x),
\]

as $\{q_n\}_{n\in\N}$ is basic. Letting $x = 0$ gives $q_n(a) = p_n(a)$. Therefore, $\{p_n\}_{n\in\N}$ is the basic set for $Q$.

The ramifications of this theorem can be distilled to the established connection between shift-invariant operators and power series. Many analogues of operations on power series can be defined for operators, a notion that will be elaborated upon in the subsequent sections.

Specifying the formula for $E^a$ in the context of the proof with $Q = D$, we derive the following expression

\begin{equation}
E^a = \sum_{k=0}^\infty a^k \frac{D^k}{k!} = e^{aD}.
\end{equation}

This identity, which dates back to Lagrange (1772) <cit.>, was written as an operator translation of Taylor's theorem. This makes the following proposition clear.

An operator is shift-invariant if and only if it commutes with the derivative $D$.

( $\imp$ ) For an operator $U$, expanding $e^{aD} U = U e^{aD}$ allows us to identify coefficients in front of $a$, revealing that $DU = UD$.

( $\isimp$ ) We generalize $DU = UD$ as $D^k U = U D^k$ for all positive integers $k$. Summing this equality with a factor of $a^k/k!$ recovers the shift-invariance.

An operator is an Appell operator if and only if it is shift-invariant and invertible.

By <Ref>, an Appell operator is characterized by $D A = A D$. The proof follows with <Ref>.

More generally than in <Ref>, the previous theorem ensures that every shift-invariant operator can be written as a power series in $D$. This enables us to apply a wide range of established results from power series theory to our operators.

If $T$ is shift-invariant, we associate a power series $\tilde T$, called the indicator, defined by $\tilde T(D) \defeq T$. It is then natural to use the composition of indicators to define a new operator

* $T \diamond U \defeq (\tilde T \circ \tilde U)(D) = \tilde T(U)$.

* For $n \in \N$, $T^{[n]} \defeq \tilde T^n(D)$. If $\tilde T$ has a compositional inverse, $n$ can also be negative.

Naturally, we have the following properties.

* $\diamond$ is associative and right-distributive over $+$.

* $D$ is the neutral element with respect to $\diamond$.

* $T^{[n]}\diamond T^{[m]} = T^{[n+m]}$.

* $(T^{[n]})^{[m]} = T^{[nm]}$.

If we look back at <Ref>, then we realize that $\tilde\Delta(t) = e^t - 1$.

Translating the fact that two power series with the same argument commute to shift-invariant operators, the subsequent corollary is concluded.

Two shift-invariant operators commute.

The expansion theorem enables us to define operator division for non-invertible shift-invariant operators. Division of operators has been extensively utilized by Steffensen [35] and others, albeit without a proper definition. We will later demonstrate that such a definition is crucial in order to avoid arriving at contradictory equations.

For $k\geq0$, let $U = D^k P$ and $V = D^k R$ be two shift-invariant operators with $R$ being invertible. Then we define

\[
U/V = \frac{U}{V} \defeq P R^{-1}.
\]

One natural property that this division still holds is the following:

\begin{equation}\label{e:(U/V)V}
\frac{U}{V} V = U.
\end{equation}

§.§ The isomorphism theorem

The isomorphism theorem is a fundamental result that is crucial to state explicitly, although it often becomes implicitly used in practice. It was first properly formulated in the work of Rota and Mullin [25], despite being extensively used by earlier authors such as Pincherle [28] and Steffensen [35]. It serves as a crucial tool for applying the known results of power series.

If $Q$ is a delta operator and $f$ a formal power series, the application $f \mapsto f(Q)$ is a ring isomorphism that maps the ring of formal power series $\tP$ to the ring of shift-invariant operators.

The proof is straightforward.

Now, we can freely apply the known properties of multiplicative and compositional inverses in the ring of formal power series to their operational analogues.

* A shift-invariant operator $T$ is invertible if and only if  $T1 \neq 0$.

* $Q$ is a delta operator if and only if $Q = D A$, for some Appell operator $A$.

* A shift-invariant operator $Q$ is a delta operator if and only if its indicator $\tilde Q$ (see <Ref>) has a power series compositional inverse $\tilde Q^{-1}$. In this case, $Q^{[-1]}$ is a well-defined delta operator.

(a) The power series $\tilde T$ has a multiplicative inverse if and only if $\tilde T(0) \neq 0$. However, $\tilde T(0) = T 1$, so by the isomorphism theorem (<Ref>), the result is proved.

(c) If $Q$ is a delta operator, then $Q 1 = 0$ and $Q x = c \neq 0$ following <Ref> and <Ref>. This is equivalent to $\tilde Q(0) = 0$ and $\tilde Q'(0) = c \neq 0$, that is $\tilde Q$ is of order $1$. This is in turn also equivalent to $\tilde Q$ having a power series compositional inverse $\tilde Q^{-1}$.

\[
Q^{[-1]} x = (\tilde Q^{-1})'(0) = \inv{\tilde Q'(\tilde Q^{-1}(0))} = \inv{\tilde Q'(0)} = \inv c \neq 0,
\]

thus $Q^{[-1]}$ is a delta operator.

$\imp$ ) Consequently, we also factor out an $x$ and write $\tilde Q(x) = x f(x)$. So by the isomorphism theorem (<Ref>), $Q = D f(D)$. Moreover, $f(0) = c \neq 0$. So by (a), $f(D)$ is an Appell operator.

( $\isimp$ ) If $Q = D A$, with $A$ an Appell operator, then $Q$ is shift-invariant and $Q x = D A x  = A 1$ a nonzero constant. So by <Ref>, $Q$ is a delta operator.

By the definition of operator division (<Ref>) and point (b) of the above proposition, we can infer the following corollary (utilizing the more concise characterization of <Ref>).

If $Q$ and $R$ are delta operators, then $Q/R$ is an Appell operator, and

\[
(Q/R)^{-1} = R/Q.
\]

The Bernoulli operator defined by

\[
\Bern \defeq \frac{D}{\Delta},
\]

is an Appell operator. Furthermore, it maps monomials to the Bernoulli polynomials

\[
\Bern x^n = \Bern_n(x).
\]

One must be cautious with operator division, as it does not behave the same as the usual division. For example, we can write $\Bern = D/(E-1)$, but we do not have the equality

\begin{equation}
\Bern = -D \sum_{k=0}^\infty E^k,
\end{equation}

as this expression diverges when applied to nonzero polynomials. However, when applied to monomials and evaluated at $0$, this gives another way of finding the regularized expressions for the Riemann zeta function at negative integers.

With all the tools developed and the notation established, we are now prepared to state the following theorem. It is significant because it demonstrates the closure of Sheffer and umbral operators under composition while providing explicit expressions for the associated delta operators in terms of power series composition. This aspect will play a central role in connecting the umbral calculus to the iterative theory in <Ref>.

If $\phi$ is a Sheffer operator for $Q$ and $\psi$ is a Sheffer operator for $R$, then

* $\psi \phi$ is a Sheffer operator for $Q \diamond R$.

* If $\phi \leadsto Q$ and $\psi \leadsto R$, then $\psi \phi \leadsto Q \diamond R$.

and for all $n\in\Z$,

* $\phi^n$ is a Sheffer operator for $Q^{[n]}$.

* If $\phi \leadsto Q$, then $\phi^n \leadsto Q^{[n]}$.

(a) According to <Ref>, $\phi D = \tilde Q(D) \phi$ and $\psi D = R \psi $. If we express $\tilde Q(t) = \sum a_k t^k$, then

\[
\psi \phi D = \psi \tilde Q(D) \phi = \sum_{k=0}^\infty a_k \psi D^k \phi = \sum_{k=0}^\infty a_k R^k \psi \phi = \tilde Q(R) \psi\phi.
\]

Using <Ref> again, we conclude that $\psi\phi$ is a Sheffer operator for $\tilde Q(R) = Q \diamond R$.

(b) Suppose that $\phi$ and $\psi$ are basic. If we write for $n>0$ $\phi_n(x) = a_1 x + \ldots + a_{n-1} x^{n-1} + a_n x^n$, then $(\psi\phi)_n(x) = \sum a_k \psi_k(x)$ and

\[
(\psi\phi)_n(0) = a_1 \psi_1(0) + \ldots + a_n \psi_n(0) = \phi_n(0) = 0.
\]

Furthermore $(\psi\phi)_0(x) = \phi_0(x) = 1$.
So, by definition, $\psi\phi$ is basic.

(c, d)  For $n\in\N$, using (a) and (b) $n$ times suffice.

For negative integers, first consider $D = \tilde Q^{-1}(Q)$ ($\tilde Q^{-1}$ exists by <Ref> (c)) and $\phi^{-1} Q = D \phi^{-1}$. $\tilde Q^{-1}$ is a power series; thus we have $\phi^{-1} D = \phi^{-1} \tilde Q^{-1}(Q) = \tilde Q^{-1} (D) \phi^{-1}$. So $\phi^{-1}$ is a Sheffer operator for $Q^{[-1]}$ by <Ref>.

If $\phi$ is basic, by <Ref>, for two variables $x,y$, $\phi_{x+y} = \phi_x \phi_y$. We can invert the formula and commute the operator to obtain $\phi_{x+y}^{-1} = \phi_x^{-1} \phi_y^{-1}$, therefore, $\phi^{-1}$ is basic. Now using (a) and (b) for $\phi^{-1}$ iteratively completes the proofs of (c) and (d).

§.§ The Pincherle derivative and the closed forms

In this section, we will develop the Pincherle derivative to facilitate the expression and proof of closed forms for basic sets (<Ref>). Subsequently, we will discuss the immediate corollaries of this theorem.

We call $T'$ the Pincherle derivative of an operator $T$, defined by

\[
T' \defeq T \x - \x T.
\]

* The Pincherle derivative is linear over polynomials of $\x$.

Let $T$ be a shift-invariant operator,

* $T'$ is shift-invariant.

* The indicator and the derivative commute, that is, $\widetilde{T'} = (\tilde T)'$

<Ref> essentially implies that the Pincherle derivative behaves similarly to the usual derivative for shift-invariant operators, with $D$ acting as a “variable”. In particular, we inherit the operational analogues of the product rule and the chain rule.

(a) is obvious. The product rule implies that $D' = D\x - \x D = 1$. With this in mind, we can easily show by induction that $(D^k)' = k D^{k-1}$, from which (b) and (c) follow.

If $\{p_n\}_{n\in\N}$ is the basic set for $Q$ then for $n\geq 0$

Transfer formula:     $p_n(x) = Q' (D/Q)^{n+1} x^n$.

Steffensen formula :     $p_n(x) = \x (D/Q)^n x^{n-1}$.

Recurrence formula :     $p_{n+1}(x) = \x (Q')^{-1} p_n(x)$.

The first pair of equations is traditionally referred to as the “transfer formulas”. To avoid ambiguity, given that both will be employed in the following, the latter shall be designated after Steffensen, in recognition of his contributions to the theorem as a whole [35].

Let us define $P \defeq D/Q$, which is well defined by <Ref> (b), and $q_n(x) = Q' P^{n+1} x^n$.
First, we establish the equivalence between the transfer formula and the Steffensen formula using the properties of the Pincherle derivative (see <Ref>). For $n \geq 0$ :

\[
Q' P^{n+1} = (DP^{-1})' P^{n+1} = (D' P^{-1} - D P'P^{-2})  P^{n+1} = P^n - P' P^{n-1} D,
\]

which we apply to the monomials

\begin{align*}
q_n(x) &= Q' P^{n+1} x^n = P^n x^n - P' P^{n-1} n x^{n-1} \\
&= P^n x^n - (P^n)' x^{n-1} = P^n x^n - (P^n \x - \x P^n) x^{n-1} \\
&=  \x P^n x^{n-1}.
\end{align*}

With this formula, we have $q_0(x) = 1$ and for $n > 0$, $q_n(0) = 0$. We want to show that $\{q_n\}_{n\in\N}$ is a Sheffer sequence for $Q = D P^{-1}$ to show its basicity. By the uniqueness of the basic sequence, we would consequently have $\{q_n\}_{n\in\N} =\{p_n\}_{n\in\N}$. To this end, we reuse the transfer formula

\[
Q q_n(x) = D P^{-1} Q' P^{n+1} x^n = Q' P^n D x^n = n Q' P^n x^{n-1} = n q_{n-1}(x).
\]

Therefore, the first two formulas hold.

For the recurrence formula, we first invert the first transfer formula. $Q'$ is invertible because $P^{-1}$ is

\[
x^n = P^{-n-1} (Q')^{-1} p_n(x).
\]

Now we apply $\x P^{n+1}$ on both sides to get the Steffensen formula back at index $n+1$ on the left hand-side

\[
p_{n+1}(x) = \x P^{n+1} P^{-n-1} (Q')^{-1} p_n(x) = \x (Q')^{-1} p_n(x).
\]

To derive the basic set $\{\varphi_n\}_{n\in\N}$ for $\Delta$, we use the recurrence relation. Given $\bra{\Delta'}^{-1} = e^{-D} = E^{-1}$, we compute

\begin{align*}
\varphi_n(x) &= \x E^{-1} \varphi_{n-1}(x) = x \varphi_{n-1}(x-1) \\
&= x (x-1) \varphi_{n-2}(x-2) = \ldots = x (x-1) \cdots 1 \cdot \varphi_0(x),
\end{align*}

$\{\varphi_n\}_{n\in\N}$ is basic, so $\varphi_0(x) = 1$, hence $\varphi_n(x) = (x)_n$ the falling factorials, as expected. Similarly, for $Q = D/\lambda$, using either of the formulas, we obtain $\str\lambda x^n = \lambda^n x^n$.

The two previous theorems exemplify the significant roles played by both polynomial sequences and their associated umbral operators. It is essential to consider both aspects in our analysis, as relying solely on one is insufficient.

The following commutation identities emerge directly from the recurrence formula.

If $\phi$ is an umbral operator for $Q$, then

\[
\phi \x = \x (Q')^{-1} \phi = \x \phi (Q^{[-1]})'.
\]

The first equality is immediate thanks to the recurrence relation of <Ref>

\[
\phi \x x^n = \phi_{n+1}(x) = \x (Q')^{-1} \phi_n(x) = \x (Q')^{-1} \phi x^n. \tag{$*$}
\]

According to <Ref>, $\phi^{-1} \leadsto Q^{[-1]}$, therefore, specifying ($*$) to $\phi^{-1}$, we obtain $\phi^{-1} \x = \x ((Q^{[-1]})')^{-1} \phi^{-1}$, which is equivalent to $\x \phi (Q^{[-1]})' = \phi \x$.

As easily as in <Ref>, we can derive the following operator identity with the first equality of <Ref>

\[
\varphi \x^n = (\x)_n E^{-n} \varphi.
\]

Furthermore, the first equality enables a characterization of basic sets through differential equations.

If $\phi$ is an umbral operator for $Q$, then $\phi_n(x)$ satisfies this differential equation

\[
\bra{n - \x \frac{Q}{Q'}}\phi_n(x) = 0
\]

It follows from using <Ref> and <Ref>

\[
n \phi_n(x) = \phi \x D x^n = \x (Q')^{-1} \phi D x^n = \x (Q')^{-1} Q \phi_n(x) = \x \frac{Q}{Q'} \phi_n(x).
\]

§.§ Extension of the results to Sheffer sets

After deriving numerous formulas for basic sets, it is natural to extend these to closely related Sheffer sets. Moreover, we will shortly present the study of cross sequences. The relation between umbral and Sheffer sets is characterized by the following proposition.

Let $Q$ be a delta operator with basic set $\{p_n\}_{n\in\N}$. Then $\{s_n\}_{n\in\N}$ is a Sheffer set for $Q$ if and only if there exists an Appell operator $A$ such that $Ap = s$.

( $\imp$ ) Let $A = s p^{-1}$. By <Ref> (a) and (c), $A$ is a Sheffer operator for $Q^{[-1]} \diamond Q = D$. Thus, by definition, $A$ is an Appell operator, and it verifies $Ap = s$.

( $\isimp$ ) Since $s = A p$ with $A$ shift-invariant, we use <Ref> twice : $Q s = A Q p = A p D = s D$, demonstrating that $s$ is a Sheffer operator for $Q$.

Let be two polynomial sequences, then these statements are equivalent

* $\{s_n\}_{n\in\N}$ is a Sheffer set with basic set $\{p_n\}_{n\in\N}$.

* For two variables $x, y$, $s_{x+y} = s_x p_y$.

* For all $x, y$ and $n\in\N$,

\[
s_n(x+y) = \sum_{k=0}^n \binom{n}{k} s_k(x) p_{n-k}(y).
\]

The equivalence between (a) and (c) was established in <cit.>.

(b $\eq$ c) The proof is analogous to that of <Ref>.

(a $\imp$ b) By <Ref>, there exists an Appell operator $A$ such that $Ap = s$. According to <Ref>, $p_{x+y} = p_x p_y$. Since $A$ is shift-invariant (by <Ref>) and using <Ref>, we have $s_{x+y} = A_{x+y}p_{x+y} = A_x p_x p_y = s_x p_y$.

(b $\imp$ a) The key is to use the associativity of the addition: on the one hand, $s_{x+y+z} = s_x p_{y+z}$, and on the other hand, $s_{x+y+z} = s_{x+y} p_z = s_x p_y p_z$. Therefore, $p_{y+z} = p_y p_z$ (because $s$ is invertible). By <Ref>, $p$ is basic, and so is $p^{-1}$ (by <Ref>). With the computation

\[
(s p^{-1})_{x+y} = s_{x+y} p^{-1}_{x+y} = s_x p_y p^{-1}_x p^{-1}_y = s_x p^{-1}_x = (s p^{-1})_x,
\]

we deduce, thanks to <Ref>, that $s p^{-1}$ is shift-invariant. Hence, by <Ref>, we conclude that $s$ has the same delta operator as the umbral operator $p$, which is equivalent to (a).

A cross operator $\phi^{(u)}$ for an umbral operator $\phi$ is a Sheffer operator (see <Ref>) defined to be of the form

\[
\phi^{(u)} = C^u \phi,
\]

for some Appell operator $C$. We call $\{\phi^{(u)}_n\}_{n\in\N}$ a cross-sequence.

Let $\phi$ be an umbral operator for $Q$. If $\phi^{(u)}$ is a cross operator for $\phi$, then it is a Sheffer operator for $Q$. Moreover, $\phi^{(u)}$ is a cross operator if and only if it verifies

\[
\phi^{(u+v)}_n(x+y) = \sum_{k=0}^n \binom{n}{k} \phi^{(u)}_k(x) \phi^{(v)}_{n-k}(y).
\]

( $\imp$ ) Since $\phi^{(v)} = C^v \phi$ is a Sheffer operator, by <Ref>,

\[
\phi^{(v)}_n(x+y) = \sum_{k=0}^n \binom{n}{k} \phi_k(x) \phi^{(v)}_{n-k}(y).
\]

Applying the shift-invariant operator $C^u$, we recover the desired identity.

( $\isimp$ ) Applying the identity at $u = v = 0$ suffices to show that $\phi \defeq \phi^{(0)}$ is an umbral operator, thanks to <Ref>. Setting $u = 0$, by <Ref>, there exists an Appell operator $F(v)$, such that $F(v) \phi = \phi^{(v)}$. We quickly deduce that $F(0)$ is the identity function. Changing $v$ to $v-1$

\[
F(v-1)\phi(x+y) = \phi^{(v-1)}_n(x+y) = \sum_{k=0}^n \binom{n}{k} \phi_k(x) \phi^{(v-1)}_{n-k}(y),
\]

now we apply $F(1)$

\[
F(1) F(v-1) \phi_n(x+y) = \sum_{k=0}^n \binom{n}{k} \phi^{(1)}_k(x) \phi^{(v-1)}_{n-k}(y) = \phi^{(v)}(x+y) = F(v) \phi_n(x+y).
\]

Therefore, $F(v) = F(1) F(v-1) = F(1)^2 F(v-2) = \ldots = F(1)^v F(0) = F(1)^v$. Looking back at the definition of $F(v)$, we conclude that $\phi^{(v)}$ is a cross operator.

It is possible to further extend the theory of cross-sequence with an analogue of Sheffer sets to basic sets named Steffensen sequences in <cit.>, but we will not develop this point here.

§ POWER SERIES ANALYSIS

We will explore various applications of umbral calculus within the study of power series. Interestingly, these findings reciprocate their utility back within umbral calculus, notably facilitated by the Isomorphism Theorem and the algebra of linear operators.

Labelle [20] first observed the profound connection between umbral calculus and iteration theory. Building on this insight, we will develop novel formulas for the composition of power series by the end of this section. Preceding this, we shall elucidate the properties of the coefficients of basic sequences, which will be the core to the mentioned subsequent analysis.

§.§ Properties of the coefficients

The coefficients of basic sequence have received little attention in the literature and have never been given proper notation. We will address this gap and demonstrate their many properties. For the sake of readability, we have adopted the Kamarata-Knuth notation style. This approach is further justified by the observation that the majority of numbers typically associated with this notation are also coefficients of basic or Sheffer sequences.

For $U \in \L(\P)$, we define its coefficients by

\[
U x^n \defeq \sum_{k\in\Z} \coeff{n}{k}_U x^k,
\]

in particular for $k < 0$, $\coeff{n}{k}_U = 0$. And, of course, if $U \in \hL$, then for $k > n$ the coefficient is also zero.

Stirling numbers of the first $\coeff{n}{k}$ and the second kind $\Stir{n}{k}$ (also denoted with the Kamarata-Knuth notation) are bound to $\varphi$ by these relationships

\begin{equation}
\coeff{n}{k}_\varphi = \coeff{n}{k} (-1)^{n-k} \quad\et\quad \coeff{n}{k}_{\varphi^{-1}} = \Stir{n}{k}.
\end{equation}

If $A$ is an Appell operator then by <Ref> (c) with $x = 0$,

\begin{equation}
\coeff{n}{k}_A = \binom{n}{k} A_{n-k}(0) = \binom{n}{k} \coeff{n-k}{0}_A.
\end{equation}

If $U \in \L(\P)$,

\begin{align*}
\coeff{n}{k}_{U \x} &= \coeff{n+1}{k}_U,& \coeff{n}{k}_{U D} &= n\coeff{n-1}{k}_U, \\[13pt]
\coeff{n}{k}_{\x U} &= \coeff{n}{k-1}_U,&  \coeff{n}{k}_{D U} &= (k+1)\coeff{n}{k+1}_U.
\end{align*}

The first key property of the coefficients is their linearity, which they inherit from the linear operators that define them.

For any operator $U, V \in \L(\P)$, and $\lambda, \mu \in \K$

\[
\coeff{n}{k}_{\lambda U + \mu V} = \lambda \coeff{n}{k}_U + \mu \coeff{n}{k}_V.
\]

[Stirling numbers recurrence relation]
Stirling numbers are famously characterized by the recurrence relation

\begin{equation}\label{e:Stir}
\coeff{n+1}{k} = n\coeff{n}{k} + \coeff{n}{k-1} \quad \et \quad \Stir{n+1}{k} = k\Stir{n}{k} + \Stir{n}{k-1}.
\end{equation}

These recurrences can be proven algebraically, but they also follow as a consequence of the identities on operators we have just established. Indeed, we can apply the second and first equalities of <Ref> to $\varphi$ and $\varphi^{-1}$. In both cases, the term $(\Delta^{[-1]})'$ can be simplified to $(1+D)^{-1}$.

Before the last step, we apply $(1+D)$ on the left

\begin{align*}
\varphi \x
&= \x \rho (\Delta^{[-1]})' = \x \varphi (1+D)^{-1} \\
&= -\varphi\x D + \x\varphi,
\end{align*}

By a direct computation
\begin{align*}
\varphi^{-1} \x
&= \x ((\Delta^{[-1]})')^{-1} \varphi^{-1} = \x (1+D) \varphi^{-1} \\
&= \x \varphi^{-1} + \x D \varphi^{-1},
\end{align*}

which are equivalent to the recurrence relations with <Ref>.

Conveniently, the second operation in the ring of linear operators, i.e. composition, also possesses an expansion.

For $n, k \in \N$ and $\phi, \psi \in \L(\P_n)$

\[
\coeff{n}{k}_{\phi\psi} = \sum_{j=k}^n \coeff{j}{k}_\phi \coeff{n}{j}_\psi.
\]

On the one hand $\phi \psi x^n = \sum_{k=0}^n \coeff{n}{k}_{\phi\psi} x^k$, while on the other hand

\[
\phi \psi x^n a= \phi \sum_{j=0}^n \coeff{n}{j}_\psi x^j = \sum_{j=0}^n \coeff{n}{j}_\psi \sum_{k=0}^j \coeff{j}{k}_\phi x^k = \sum_{0 \leq k \leq j \leq n} \coeff{j}{k}_\phi \coeff{n}{j}_\psi x^k = \sum_{k=0}^n \bra{\sum_{j = k}^n \coeff{j}{k}_\phi \coeff{n}{j}_\psi} x^k.
\]

Thus, by identifying the coefficients, we get the desired result.

The “connection constant problem” for two basic sequences $\{\phi_n\}_{n\in\N}$ and $\{\psi_n\}_{n\in\N}$ can be solved by finding the coefficients of $\psi^{-1} \phi$, because

\[
\phi_n(x) = \sum_{k=0}^n \coeff{n}{k}_{\psi^{-1}\phi} \psi_k(x).
\]

The formula becomes obvious upon application of $\psi^{-1}$. The coefficients can thus be computed by analyzing $\psi^{-1} \phi$, which is an umbral operator thanks to <Ref>, or by separately analyzing $\psi^{-1}$ and $\phi$, and using <Ref>.

A special and useful corollary for iteration is the following.

For $n, k \in \N$, $p, q \in \Z$ and $\phi \in \hL$

\[
\coeff{n}{k}_{\phi^{p + q}} = \sum_{j=k}^n \coeff{j}{k}_{\phi^p} \coeff{n}{j}_{\phi^q}.
\]

In one of his exercises, Knuth <cit.> asked to prove that the binomial transform he introduced was an involution. This fact, also widely known among French students and mathematicians as Pascal's inversion formula, has probably been known for centuries. There is an analogous invertible transform using Stirling numbers known as the Stirling transform [5], which also appears in Knuth's work <cit.>. Using <Ref>, we demonstrate not only that these transforms are part of a broad family of invertible transforms <cit.>, but also that they have a dual invertible transform.

Let $\phi\in\hL$ and $(a_i)_i, (b_i)_i$ be tuples of an abelian group with internal law $+$.

For $n \in \N$ and $0 \leq m \leq n$

\[
\sum_{k=m}^n \coeff{n}{k}_\phi b_k = a_n \eq b_n = \sum_{k=m}^n \coeff{n}{k}_{\phi^{-1}} a_k.
\]

For $k \in \N$ and $k \leq j \leq \infty$

\[
\sum_{n=k}^j \coeff{n}{k}_\phi b_n = a_k \eq b_k = \sum_{n=k}^j \coeff{n}{k}_{\phi^{-1}} a_n.
\]

By replacing $b_i$ in the formula of $a_i$ with its expression

\begin{align*}
&\sum_{j=m}^n \coeff{n}{j}_\phi \sum_{k=m}^j \coeff{j}{k}_{\phi^{-1}} a_k \\
&= \sum_{m\leq k \leq j\leq n} \coeff{n}{j}_\phi \coeff{j}{k}_{\phi^{-1}} a_k \\
&= \sum_{k=m}^n a_k \sum_{j=k}^n  \coeff{n}{j}_\phi \coeff{j}{k}_{\phi^{-1}} \\
&= \sum_{k=m}^n a_k \delta_{n,k} = a_n,
\end{align*}

\begin{align*}
&\sum_{m=k}^j \coeff{m}{k}_\phi \sum_{n=m}^j \coeff{n}{m}_{\phi^{-1}} a_n\\
&= \sum_{k\leq m \leq n\leq j} \coeff{m}{k}_\phi \coeff{n}{m}_{\phi^{-1}} a_n \\
&= \sum_{n=k}^j a_n \sum_{m=k}^n  \coeff{m}{k}_\phi \coeff{n}{m}_{\phi^{-1}} \\
&= \sum_{n=k}^j a_n \delta_{k, n} = a_k,
\end{align*}

where we have used <Ref> in the last steps.

$\phi$ is an umbral operator if and only if

\[
\binom{i+j}{i} \coeff{n}{i+j}_{\phi} = \sum_{k=0}^n \binom{n}{k} \coeff{k}{i}_\phi \coeff{n-k}{j}_\phi.
\]

Umbral operators are characterized by their associated polynomial sequence being of binomial type (<Ref>). The binomial type property can, in turn, be characterized by this identity on the coefficients, specifically through an identification of the coefficient in front of $x^i y^j$ in <ref>.

To conclude this section, we want to address a simple yet surprising corollary of the Steffensen formula. It is known that deriving the coefficients of integer exponents in power series presents considerable challenges. The following proposition provides a formula for a finite set of these coefficients in terms of the coefficients of an umbral operator.

Let $\phi \leadsto Q$, $A = D/Q$ and $\tilde A(t)^n \defeq \sum a_k^{(n)} \frac{t^k}{k!}$ then for $1\leq k \leq n$

\[
a_{n-k}^{(n)} =\coeff{n}{k}_\phi\binom{n-1}{k-1}^{-1}.
\]

Using the definition of the coefficients and the Steffensen formula of <Ref>, we obtain

\[
\sum_{k=0}^n \coeff{n}{k}_\phi x^n = \phi_n(x) = \x A^n x^{n-1}
= \x \sum_{k=0}^\infty a_k^{(n)} \frac{D^k}{k!} x^{n-1} = \sum_{k=0}^{n-1} a_k^{(n)} \binom{n-1}{k} x^{n-k},
\]

from which we can set $j = n-k$ in the right sum to identify the coefficients.

For $\varphi \leadsto \Delta$, $A = \Bern$, the Bernoulli operator specified in <Ref>. The generalized Bernoulli numbers are defined by

\begin{equation}
\tilde\Bern(t)^n = \bra{\frac{t}{e^t - 1}}^n = \sum_{k=0}^\infty \Bern_k^{(n)} \frac{t^k}{k!},
\end{equation}

extending the Bernoulli numbers $\Bern_k \defeq \Bern_k(0) = \Bern^{(1)}_k$. By <Ref>, the generalized one can be expressed in terms of Stirling numbers, defined in <Ref>, for $0 \leq k < n$

\begin{equation}
\Bern_k^{(n)} = \coeff{n}{n-k}_\varphi \binom{n-1}{n-k-1}^{-1} = (-1)^k \coeff{n}{n-k} \binom{n-1}{k}^{-1}.
\end{equation}

This is equation (2.10) of [24].

§.§ The generating functions

In this section, we briefly study the generating functions of the basic sequences and their coefficients. Although it is known that generating functions alone are not sufficient to build umbral calculus [30]. This insufficiency is exemplified by the absence, in Sheffer's work [33], of certain results derived by Steffensen <Ref>. Nevertheless, their study provides a very powerful bridge to power series theory.

We define the bivariate exponential generating function of $\phi \in \hL$ and its coefficients by

\[
\G_\phi(x, t) \defeq \phi e^{xt} = \sum_{n=0}^\infty \phi_n(x) \frac{t^n}{n!}, \qquad \g_\phi (k, t) \defeq \sum_{n=k}^\infty \coeff{n}{k}_\phi \frac{t^n}{n!}.
\]

\[
[t^n]\G_\phi(x,t) = \frac{\phi_n(x)}{n!}, \qquad [x^k]\G_\phi(x,t) = \g_\phi(k, t).
\]

The first equality is a straightforward restatement of the definition. For the second, we simply exchange the summations

\[
\G_\phi(x, t) = \sum_{n=0}^\infty \frac{t^n}{n!} \phi_n(x) = \sum_{n=0}^\infty \frac{t^n}{n!} \sum_{k=0}^n \coeff{n}{k}_\phi x^k = \sum_{0 \leq k \leq n < \infty} \coeff{n}{k}_\phi \frac{t^n}{n!} x^k =\sum_{k=0}^\infty \g_\phi(k, t) x^k.
\]

Let $\phi \leadsto Q$ and $A$ be an Appell operator. $s = A \phi$ if and only if

\[
\G_s(x, t) = \tilde A(\tilde Q^{-1}(t)) e^{x\tilde Q^{-1}(t)}.
\]

We will prove the equivalent statement

\[
G_s(a, Q) = A e^{a D} = A E^a, \tag{$\ast$}
\]

which follows from the evaluation at $\tilde Q(t)$ and the isomorphism theorem.

( $\imp$ ) By the expansion theorem (<Ref>)

\[
E^a A = \sum_{k=0}^\infty (\E E^a A \phi_k(x)) \frac{Q^k}{k!} = \sum_{k=0}^\infty (\Ev{a} s_k(x)) \frac{Q^k}{k!} = \sum_{k=0}^\infty s_k(a) \frac{Q^k}{k!} = \G_s(a, Q).
\]

( $\isimp$ ) We apply $(*)$ to $\phi_n(x)$

\[
A \phi_n(x+a) = \sum_{k=0}^n \binom{n}{k} s_k(a) \phi_{n-k}(x),
\]

but we also have

\[
A \phi_n(x+a) = A_a \sum_{k=0}^n \binom{n}{k} \phi_k(a) \phi_{n-k}(x).
\]

By identification, we get $s_k(a) = A_a \phi_k(a)$, thus

\[
A \phi_n(x+a) = s_n(x+a) = \sum_{k=0}^n \binom{n}{k} s_k(a) \phi_{n-k}(x),
\]

and using <Ref> concludes the proof.

If we identify the coefficients in front $x^k$ in the formula from <Ref> through <Ref>, we deduce this corollary.

\[
\g_s(k,t) =  \tilde A(\tilde Q^{-1}(t)) \frac{\tilde Q^{-1}(t)^k}{k!}.
\]

Using $A = 1$, we specialize the two previous results to basic sequences.

If $\phi \leadsto Q$,

\[
\G_\phi(x, t) = e^{x \tilde Q^{-1}(t)} \quad\et\quad \g_\phi(k, t) = \frac{\tilde Q^{-1}(t)^k}{k!}.
\]

The generating functions provide a surprising tool for computing the coefficients of an umbral operator.

If $\phi \leadsto Q$, then the following equality holds

\[
\coeff{n}{k}_\phi = \coeff{n}{k}_{(Q^{[-1]}/D)^k}.
\]

The formula finds its main use when $Q^{[-1]}$ has a more convenient form than $Q$ and behaves well under exponentiation.

By the isomorphism theorem applied to <Ref>

\[
\bra{\frac{Q^{[-1]}}{D}}^k = k! \frac{\g_\phi(k, D)}{D^k} = k! \sum_{p=k}^\infty \coeff{p}{k}_{\phi} \frac{D^{p-k}}{p!} = k! \sum_{p=0}^\infty \coeff{p+k}{k}_{\phi} \frac{D^p}{(p+k)!}.
\]

We apply the equality to the monomials

\[
\bra{\frac{Q^{[-1]}}{D}}^k x^n = k! \sum_{p=0}^n \coeff{p+k}{k}_{\phi} \frac{n!x^{n-p}}{(n-p)!(p+k)!},
\]

and identify the $k$-th coefficient

\[
\coeff{n}{k}_{(Q^{[-1]}/D)^k} = k! \coeff{n-k+k}{k}_\phi \frac{n!}{k!(n-k+k)!} = \coeff{n}{k}_\phi.
\]

In [8], it was observed that the multivariate generalization of the umbral calculus made clear the equivalence between the Lagrange inversion formula and the Steffensen formula. We will briefly demonstrate this connection, leveraging <Ref> as an intermediate result bridging the two.

Let $f$ be an invertible power series. Then the following formula holds for $k \in \N^*$

\[
f^{-1}(t)^k = k\sum_{n=k}^\infty \frac{t^n}{n} [x^{n-k}] \bra{\frac{x}{f(x)}}^n,
\]

With <Ref>, we can write $x/f(x)= A(x)$. Let $\phi$ be the delta operator for $f(D)$. Using the notation and the result of <Ref>, we get

\[
[x^{n-k}] A(x)^n = \frac{a^{(n)}_{n-k}}{(n-k)!} = \frac{\coeff{n}{k}_\phi}{(n-k)!\binom{n-1}{k-1}} = \frac{(k-1)!}{(n-1)!}\coeff{n}{k}_\phi,
\]

thus, with the help of <Ref>

\[
k\sum_{n=k}^\infty \frac{t^n}{n} [x^{n-k}] A(x)^n = k!\sum_{n=k}^\infty \frac{t^n}{n!} \coeff{n}{k}_\phi = k! \g_\phi(k, t) = f^{-1}(t)^k,
\]

Equivalently, we can rewrite the formula as

\[
f^{-1}(t)^k = \sum_{n=1}^\infty \frac{t^n}{n} [x^{n-1}] kx^{k-1} \bra{\frac{x}{f(x)}}^n.
\]

Summing it over $k$ with the coefficients of some power series $\Phi$ gives, by linearity, Bürman's formulation of the result

\begin{equation}
\Phi(f^{-1}(t)) = \sum_{n=1}^\infty \frac{t^n}{n} [x^{n-1}] \Phi'(x)\bra{\frac{x}{f(x)}}^n.
\end{equation}

In [27], Niederhausen states a beautiful umbral calculus theorem which was surprisingly not known prior. Only a close equivalent statement was found not long before in the proof of an identity for Bell polynomials (the subject of the next section) <cit.>.

If $\phi \leadsto Q$ then we define $\psi$ through its coefficients

\begin{equation}\label{e:Nieder}
\coeff{n}{k}_\psi = \binom{n}{k} \phi_{n-k}(k),
\end{equation}

for $0 \leq k \leq n \in \N$. $\psi$ is an umbral operator for $R$ and characterized by

\begin{equation}
R^{[-1]} = D e^{Q^{[-1]}} \quad\ou\quad R^k = k \sum_{n=k}^\infty \frac{\phi_{n-k}(-n)}{n(n-k)!} D^n.
\end{equation}

In particular, for $k=1$, the delta operator $R$ is given by

\begin{equation}
R = \sum_{n=1}^\infty \frac{\phi_{n-1}(-n)}{n!} D^n.
\end{equation}

The inverse Niederhausen transform is meaningful whenever, given a $Q$, $\log(Q^{[-1]}/D)$ forms a delta operator. This is achieved when $Q^{[-1]} = D + cD^2 + \ldots$ with $c\neq0$.  In loose terms and under these conditions, the inverse transform asserts that the coefficients of every basic set are also basic sets. This fact was hinted at by the long-known characterization of umbral operators of <Ref>.

Rewriting the the <ref> as

\[
\coeff{n+k}{k}_\psi \frac{t^{n+k}}{(n+k)!} = \inv{k!} \phi_n(k) \frac{t^{n+k}}{n!},
\]

and summing it over $\N$ yields for the left hand-side

\[
\sum_{n=0}^\infty \coeff{n+k}{k}_\psi \frac{t^{n+k}}{(n+k)!} = \sum_{n=k}^\infty \coeff{n}{k}_\psi \frac{t^n}{n!} = \g_\psi(k, t),
\]

while for the right hand-side

\[
\inv{k!} \sum_{n=0}^\infty \phi_n(k) \frac{t^{n+k}}{n!} = \frac{t^k \G_\phi(k, t)}{k!} = \frac{t^k e^{k \tilde Q^{-1}(t)}}{k!}.
\]

We reached the equality

\[
\g_\psi(k, t) = \frac{(te^{\tilde Q^{-1}(t)})^k}{k!},
\]

thus by <Ref>, we see that $\psi$ is basic for $R$ defined by $R^{[-1]} = D e^{Q^{[-1]}}$.
Furthermore, using Lagrange inversion and <Ref>

\begin{align*}
R^k &= k \sum_{n=k}^\infty \frac{D^n}{n} [x^{n-k}] \bra{\frac{x}{xe^{\tilde Q^{-1}(x)}}}^n = k \sum_{n=k}^\infty \frac{D^n}{n} [x^{n-k}] e^{-n\tilde Q^{-1}(x)} \\
&= k \sum_{n=k}^\infty \frac{D^n}{n} [x^{n-k}] G_\phi(-n, x) = k \sum_{n=k}^\infty \frac{D^n}{n}\frac{\phi_{n-k}(-n)}{(n-k)!}.
\end{align*}

§.§ Bell polynomials

Here and in <Ref>, $f$ denotes a formal power series of order 1. We write $f$ as

\[
f(x) = \sum_{k=1}^\infty \frac{a_k}{k!} x^k, \quad a_1 \neq 0.
\]

We say $f$ is unitary, when $f'(0) = a_1 = 1$. We know, thanks to <Ref> (c), that $Q \defeq f^{-1}(D)$ is a well-defined delta operator, and it is not hard to work out that for $s\in\Z$, $Q^{[s]} x = a_1^s$ because $(f^s)'(0) = a_1^s$. Therefore, if $f$ is unitary, so is $f^s$ and $Q^{[s]}$. In addition, if we define $\phi\leadsto Q$, then $\phi^s = Q^{[s]}$, which means $\phi^s$ is also unitary, by <Ref>.

Partial and Complete Bell polynomials <cit.> are respectively defined for $k\in\N$ by

\begin{equation}\label{e:pBelldef}
\frac{f(x)^k}{k!} = \sum_{n=k}^\infty B_{n,k}(a_1, \ldots, a_{n-k+1}) x^n,
\end{equation}

\begin{equation}\label{e:Belldef}
e^{f(t)}= \sum_{n=0}^\infty B_n(a_1, \ldots, a_n) t^n.
\end{equation}

Both kind are linked through the relationship

\begin{equation}
B_n(a_1, \ldots, a_n) = \sum_{k=0}^n B_{n,k}(a_1, \ldots, a_{n-k+1}).
\end{equation}

which can be proved by summing over integers $k$ <ref>. We can also identify the coefficients in <ref> to obtain the following formulas

\begin{equation}
B_n(a_1, \ldots, a_n) =  n! \sum_{1i_1 + \ldots + ni_n = n} \prod_{p=1}^n \frac{a_p^{i_p}}{(p!)^{i_p} i_p!},
\end{equation}

\begin{equation}
B_{n,k}(a_1, \ldots, a_{n-k+1}) = n! \sum_{\substack{1i_1 + \ldots + ni_n = n\\ i_1 + \ldots + i_n = k}} \prod_{p=1}^{n-k+1} \frac{a_p^{i_p}}{(p!)^{i_p} i_p!}.
\end{equation}

It is known that Bell polynomials are closely related to basic sequences. Indeed, by <Ref> we recover

\begin{equation}
\phi_n(x) = B_n(a_1 x, \ldots, a_n x),
\end{equation}

\begin{equation}\label{e:coeffBell}
\coeff{n}{k}_\phi = B_{n,k}(a_1, \ldots, a_{n-k+1}).
\end{equation}

Thus, the coefficients $a_n = \coeff{n}{1}_\phi = [x]\phi_n(x)$ uniquely determine all of the higher order coefficients of the polynomial. The same can be said about the coefficient of powers of $\phi$. Indeed by <Ref>, for any integer $s$, $\phi^s \leadsto f^{-s}(D)$. Thus we are easily able to find the following formula

\begin{equation}\label{e:fsk}
\frac{f^s(t)^k}{k!} = \sum_{n=k}^\infty \coeff{n}{k}_{\phi^s} \frac{t^n}{n!},
\end{equation}

where the coefficients can be expressed by repeated use of <Ref>

\begin{align*}
\coeff{n}{k}_{\phi^s} &= \sum_{j_1=k}^n \coeff{j_1}{k}_\phi \coeff{n}{j_1}_{\phi^{s-1}} = \sum_{j_1=k}^n \coeff{j_1}{k}_\phi \sum_{j_2=j_1}^n \coeff{j_2}{j_1}_\phi \coeff{n}{j_2}_{\phi^{s-2}} \\
&= \sum_{k=j_0\leq j_1 \leq j_2 \leq j_3 = n} \coeff{j_1}{j_0}_\phi \coeff{j_2}{j_1}_\phi \coeff{j_3}{j_2}_{\phi^{s-2}},
\end{align*}

continuing the expansion, we arrive at

\begin{equation}\label{e:phi^s}
\coeff{n}{k}_{\phi^s} = \sum_{k=j_0 \leq \ldots \leq j_s = n} \prod_{i=0}^{s-1} \coeff{j_{i+1}}{j_i}_\phi,
\end{equation}

and finally, we can use <ref> to obtain

\begin{equation}\label{e:compo}
\coeff{n}{k}_{\phi^s} = \sum_{k=j_0 \leq \ldots \leq j_s = n} \prod_{i=0}^{s-1} B_{j_{i+1}, j_i}(a_1, \ldots, a_{j_{i+1}-j_i+1}).
\end{equation}

This formula also provides a new method for determining the coefficients of compositional iterates of an invertible power series. To our knowledge, it is new.

One further observation is that Jabotinsky matrices are given precisely by $(\frac{k!}{n!}\coeff{n}{k}_\phi)_{n,k\in\N}$. However, what makes this new notation strictly better is that a matrix analogue of <Ref>, though very natural in the present theory, has never been derived, thereby explaining the absence of <ref> in the literature.

§.§ Application to iteration theory

Umbral calculus, as we have constructed it, can be interpreted as an investigation into the structure of invertible power series through the following series of bijections :

\[
f \longleftrightarrow Q \longleftrightarrow \phi \longleftrightarrow \{\phi_n\}_{n\in\N} \longleftrightarrow \bra{\coeff{n}{k}_\phi}_{n,k\in\N},
\]

with $Q = f^{-1}(D)$, where $f$ was previously defined, and $\phi \leadsto Q$. The initial bijection integrates the inversion of $f$ with the isomorphism delineated in <Ref>, while the subsequent bijection arises from <Ref> (b). The third bijection corresponds to <Ref>, and the final bijection links the basic sequence to its coefficients. By virtue of these bijections, it is understood that these new objects encapsulate information equivalent to that contained in $f$ itself, while offering greater ease of manipulation. Their properties facilitate more straightforward computations of complex formulas associated with $f$. This was demonstrated, for example, in the quick proof of Lagrange's inversion (<Ref>), or in the new formula of <ref>. The objective of this section is to showcase the effectiveness of the previously developed tools in generalizing <ref> to fractional iteration. It is motivated by the insights provided by Labelle [20] in first drawing a connection between umbral calculus and iteration theory.

A notable milestone in iteration theory was due to Écalle [37, 38], in his use of the composition operator $C_f$ defined by

\begin{equation}
C_f g(x) \defeq g(f(x)).
\end{equation}

For example, if $f(x) = \lambda x$, then $C_f = \str\lambda$. The usual composition operation $\circ$ lacks basic properties (such as left-distributivity) that makes it unusable. On the other hand, $C_f$ is a linear operator satisfying the obvious

\begin{equation}
C_f C_g = C_{g \circ f},
\end{equation}

and for any integer $p$

\begin{equation}
(C_f)^p = C_{f^p}.
\end{equation}

With this operator at hand, Écalle simplified many proofs of formulas in iterative theory due to Jabotinsky [16, 17].

First, we define fractional iteration (justified in <cit.>) as well as central related ideas.

We define complex iteration by $f^1 = f$ and for all $r, s \in \K$,

\[
f^r \circ f^s = f^{r+s}.
\]

Also, we define the iterative logarithm $\itlog$ of $f$ ($f_*$ for short) by

\[
\itlog(f) \defeq \left.\odv{f^s}{s}\right|_{s=0},
\]

and the iterative exponential $\itexp$ of $f$ by

\[
\itexp(f) \defeq e^{f(\x)D},
\]

which is a linear operator.

If $f$ is unitary, then for $r, s \in \K$

* $(f^r)^s = f^{rs}$.

* $\displaystyle f_*(\x) D = \sum_{p=1}^\infty \frac{(-1)^{p-1}}{p} (C_f - 1)^p$.

* $\displaystyle C_{f^s} = \sum_{p=0}^\infty \binom{s}{p} (C_f - 1)^p$.

* $C_{f^s} = \itexp(s \cdot \itlog(f))$.

All of these properties quickly follow from a well-orchestrated proof. For some power series $g$, first we compute via the chain rule

\begin{align*}
\odv{}{s} g(f^s(x)) &= \left.\odv{}{s} g(f^{s+t}(x))\right|_{s=0} = \left.\odv{}{t} g(f^s(f^t(x)))\right|_{t=0} = \left.\odv{f^t(x)}{t} \right|_{t=0} \odv{g(f^s(x))}{x} \\
&= f_*(x) \odv{g(f^s(x))}{x} = f_*(\x) D g(f^s(x)).
\end{align*}

This constitutes a differential equation in $s$ with solution

\[
g(f^s(x)) = e^{s\itlog(f)(\x) D} g(x),
\]

from which we can identify equation (d), while the following calculation let us recover equation (c)

\[
C_{f^s} = e^{s\itlog(f)(\x) D} = (e^{\itlog(f)(\x) D})^s = (C_f)^s = \sum_{p=0}^\infty \binom{s}{p} (C_f - 1)^p.
\]

Therefore, we can also perform this computation

\[
C_{(f^r)^s} = (C_{f^r})^s = ((C_f)^r)^s = (C_f)^{rs} = C_{f^{rs}},
\]

where the third equality is justified by <cit.>. Applying this to the identity function, let us retrieve (a). For equation (b), we simply differentiate equation (d) at $s=0$

\[
f_*(\x) D = \left. \odv{C_{f^s}}{s} \right|_{s=0} = \sum_{p=0}^\infty \left.\odv{}{s} \binom{s}{p}\right|_{s=0} (C_f - 1)^p = \sum_{p=1}^\infty \frac{(-1)^{p-1}}{p} (C_f - 1)^p.
\]

We can apply (b) to the identity function and evaluating the result at  $x\in\K$ to obtain

\[
f_*(x) = \sum_{p=1}^\infty \frac{(-1)^{p-1}}{p} \sum_{\ell=0}^p \binom{p}{\ell} (-1)^{p-\ell} f^{\ell}(x),
\]

and we can use <ref> to isolate the coefficients of the power series expansion of $f_*$

\[
f_*(x) = \sum_{n=1}^\infty \frac{x^n}{n!} \sum_{p=1}^\infty \frac{(-1)^{p-1}}{p} \sum_{\ell=0}^p \binom{p}{\ell} (-1)^{p-\ell} \coeff{n}{1}_{\phi^\ell}.
\]

The expression we find can be further simplified by means of the following lemma.

If $\psi$ is invertible and unitary, and $p \in \N$ then

\[
\coeff{n}{k}_{(\psi-1)^p} = \sum_{\ell=0}^p \binom{p}{\ell} (-1)^{p-\ell} \coeff{n}{k}_{\psi^\ell} = \sum_{k=j_0 < \ldots < j_p = n} \prod_{i=0}^{p-1} \coeff{j_{i+1}}{j_i}_\psi,
\]

and if $p > n-k$, it is also equal to $0$.

On the one hand, by we make use of the binomial theorem and the linearity of the coefficient (<Ref>)

\[
\coeff{n}{k}_{(\psi-1)^p} = \coeff{n}{k}_{\sum_{\ell=0}^p \binom{p}{\ell}(-1)^{p-\ell} \psi^\ell} = \sum_{\ell=0}^p \binom{p}{\ell} (-1)^{p-\ell} \coeff{n}{k}_{\psi^\ell},
\]

while on the other hand, we can safely use <ref>, to obtain

\[
\coeff{n}{k}_{(\psi-1)^p} = \sum_{k=j_0 \leq \ldots \leq j_p = n} \prod_{i=0}^{p-1} \coeff{j_{i+1}}{j_i}_{\psi-1}.
\]

Because $(\psi-1)_n(x) = \psi_n(x) - x^n$ and $\psi$ are unitary, $\coeff{n}{n}_{\psi-1} = 0$ and for $k < n$, $\coeff{n}{k}_{\psi-1} = \coeff{n}{k}_{\psi}$. Therefore, we see that the summation is done over some zeros corresponding to the cases where $j_i = j_{i+1}$ for some $i$, hence

\[
\coeff{n}{k}_{(\psi-1)^p} = \sum_{k=j_0 < \ldots < j_p = n} \prod_{i=0}^{p-1} \coeff{j_{i+1}}{j_i}_\psi.
\]

In this formula, we notice that if $p > n-k$, the summation must be empty because there are more $j_i$ indices than distinct integers between $k$ and $n$.

The coefficients of an umbral operator are, in fact, except for the convention of the coefficients of $f$, exactly the same as the coefficients of the Jabotinski matrices. However, the properties we proved in <Ref>, makes the coefficients of umbral operators strictly more powerful to manipulate. An example of why is the following theorem, which was proved by Jabotinski [17], but no analog of <ref> nor <Ref> was given to compute the coefficients of his matrices.

If $f$ is unitary

\begin{align*}
\frac{f^s(x)^k}{k!}
&= \sum_{n=k}^\infty \frac{x^n}{n!} \sum_{p=0}^{n-k} \binom{s}{p} \coeff{n}{k}_{(\phi-1)^p} \\
&= \sum_{n=k}^\infty \frac{x^n}{n!} \sum_{p=0}^{n-k} \binom{s}{p} \binom{n-k-s}{n-k-p} \coeff{n}{k}_{\phi^p}
\end{align*}

\begin{align*}
&= \sum_{n=2}^\infty \frac{x^n}{n!} \sum_{p=1}^{n-1} \frac{(-1)^{p-1}}{p} \coeff{n}{1}_{(\phi-1)^p} \\
&= \sum_{n=2}^\infty \frac{x^n}{n!} \sum_{p=1}^{n-1} \frac{(-1)^{p-1}}{p} \binom{n-1}{p} \coeff{n}{1}_{\phi^p}.
\end{align*}

The reader must be careful and should not identify what differs in the formulas as they are not equal.

Just as in <Ref>, we retrieve these formulas with <Ref> (d) applied to the identity and (b) to the function $x \mapsto x^k/k!$

\[
\begin{cases}
\displaystyle f_*(x) = \sum_{n=1}^\infty \frac{x^n}{n!} \sum_{p=1}^\infty \frac{(-1)^{p-1}}{p} \sum_{\ell=0}^p \binom{p}{\ell} (-1)^{p-\ell} \coeff{n}{1}_{\phi^\ell} \\
\displaystyle\frac{f^s(x)^k}{k!} = \sum_{n=1}^\infty \frac{x^n}{n!} \sum_{p=0}^\infty \binom{s}{p} \sum_{\ell=0}^p \binom{p}{\ell} (-1)^{p-\ell} \coeff{n}{k}_{\phi^\ell}.
\end{cases}
\]

The ranges of the sums can be restricted thanks to <Ref>

\[
\begin{cases}
\displaystyle f_*(x) = \sum_{n=2}^\infty \frac{x^n}{n!} \sum_{p=1}^{n-1} \frac{(-1)^{p-1}}{p} \sum_{\ell=0}^p \binom{p}{\ell} (-1)^{p-\ell} \coeff{n}{1}_{\phi^\ell} \\
\displaystyle\frac{f^s(x)^k}{k!} = \sum_{n=1}^\infty \frac{x^n}{n!} \sum_{p=0}^{n-k} \binom{s}{p} \sum_{\ell=0}^p \binom{p}{\ell} (-1)^{p-\ell} \coeff{n}{k}_{\phi^\ell}.
\end{cases}
\]

<Ref> also gives the first formulas we wish to prove. For the second formula, its simply a matter of exchanging the sums and using known properties for binomial coefficients. We will refer to the properties of <cit.> with quotes.

For $f_*$, first notice that in the sum over $\ell$, the term at $\ell = 0$ is always $0$ because $\coeff{n}{1}_{\phi^0} = \delta_{n,1}$ and $n \geq 2$. We now use the “absorption/extraction” property

\begin{align*}
&\sum_{p=1}^{n-1} \frac{(-1)^{p-1}}{p} \sum_{\ell=1}^p \binom{p}{\ell} (-1)^{p-\ell} \coeff{n}{1}_{\phi^\ell} = \sum_{1\leq\ell\leq p<n} \inv{\ell} \binom{p-1}{\ell-1} (-1)^{-\ell-1} \coeff{n}{1}_{\phi^\ell} \\
={}& \sum_{\ell=1}^n \frac{(-1)^{\ell-1}}{\ell} \coeff{n}{1}_{\phi^\ell} \sum_{p=\ell}^{n-1} \binom{p-1}{\ell-1}
= \sum_{\ell=1}^n \frac{(-1)^{\ell-1}}{\ell} \coeff{n}{1}_{\phi^\ell} \sum_{p=\ell-1}^{n-2} \binom{p}{\ell-1}.
\end{align*}

The desired result follows upon application of the “upper summation”

\[
\sum_{p=\ell-1}^{n-2} \binom{p}{\ell-1} = \binom{n-1}{\ell}.
\]

For the final formula, we shall use the “trinomial revision” identity

\begin{align*}
&\sum_{p=0}^{n-k} \binom{s}{p} \sum_{\ell=0}^p \binom{p}{\ell} (-1)^{p-\ell} \coeff{n}{k}_{\phi^\ell} = \sum_{0\leq\ell\leq p\leq n-k} \binom{s}{\ell} \binom{s-\ell}{p-\ell} (-1)^{p-\ell} \coeff{n}{k}_{\phi^\ell} \\
={}& \sum_{\ell=0}^{n-k} \binom{s}{\ell} \coeff{n}{k}_{\phi^\ell} \sum_{p=\ell}^{n-k} \binom{s-\ell}{p-\ell} (-1)^{p-\ell} = \sum_{\ell=0}^{n-k} \binom{s}{\ell} \coeff{n}{k}_{\phi^\ell} \sum_{p=0}^{n-k-\ell} \binom{s-\ell}{p} (-1)^p,
\end{align*}

and we end the proof by first applying the “upper negation” and then the “parallel summation”

\begin{align*}
\sum_{p=0}^{n-k-\ell} \binom{s-\ell}{p} (-1)^p = \sum_{p=0}^{n-k-\ell} \binom{p - (s - \ell) -1}{p} = \binom{n-k-s}{n-k-\ell}.
\end{align*}

The case $k=1$ and $s\in\N$ of the first equality of <Ref> rewritten with <Ref> as

\begin{equation}
\frac{f^s(x)^k}{k!} = \sum_{n=k}^\infty \frac{x^n}{n!} \sum_{p=0}^{n-k} \binom{s}{p} \sum_{k=j_0 < \ldots < j_p = n} \prod_{i=0}^{p-1} \coeff{j_{i+1}}{j_i}_\phi,
\end{equation}

was known to Schröder <cit.>. <Ref> is different from Schröder's formula and can be interpreted as a simplification of it when $s\in\N$.

Koszul numbers $K_n$ are the name (proposed in [22]) for the coefficient of the iterative logarithm of $\tilde \Delta(t) = e^t-1$:

\[
\itlog(\tilde \Delta)(x) = \sum_{n=2}^\infty \frac{K_n}{n!} x^n.
\]

The formula for $K_n$ is given by <Ref>. The formula can be simplified by the identity $\Stir{n}{1} = 1$ for $n > 0$, which is a consequence of the recurrence relation of <Ref>. Thus, we arrive at

\[
K_n = \sum_{p=1}^{n-1} \frac{(-1)^{p-1}}{p} \sum_{1 < j_1 < \ldots < j_p = n} \prod_{i=1}^{p-1} \Stir{j_{i+1}}{j_i}.
\]

This formula was first proved (with difficulty) by Aschenbrenner [3], in the process of showing a conjecture, and inspired the result of <Ref>.

§ THE OPERATOR POINT OF VIEW

This section presents a collection of various results related to operators. We begin by proving both established and novel identities involving operators, which are then used to characterize a special class of umbral operators. We also derive explicit formulas for general umbral operators and construct pseudoinverses of delta operators.

§.§ A sidestep into operational calculus

The expansion theorem provides a formula for expanding shift-invariant operators in terms of the derivative $D$. When considering non shift-invariant operators, given that the multiplication operator $\x$ is the simplest of them, an inquiry arises as to whether an analogous expansion in terms of $\x$ and $D$ exists, and, if so, the specific formulation of such an expansion. The first question was addressed by Pincherle and Amaldi <cit.> and a definitive resolution was derived much later by Kurbanov and Maksimov [19]. Di Bucchianico and Loeb [6] shedding light on this previously unknown contribution, generalized it with delta operators. In the following, we further extend their theorem in a way that was, in fact, tacitly used, but unstated, in their Proposition 21 (our <Ref>).

If $U \in \L(P)$, $\phi \leadsto Q$ and

\[
\frac{\G_{U \phi}(x, t)}{\G_\phi(x, t)} = \sum_{k=0}^\infty f_k(x) g_k(t),
\]

for some formal power series $\{f_k\}_{k\in\N}$ and $\{g_k\}_{k\in\N}$, then $U$ can be written as

\[
U = \sum_{k=0}^\infty f_k(\x) g_k(Q).
\]

By construction

\[
\sum_{k=0}^\infty f_k(\x) g_k(Q) \G_\phi(x, t) = \sum_{k=0}^\infty f_k(x) g_k(t) \G_\phi(x, t) = \G_{U \phi}(x, t).
\]

After identifying the coefficients of $t^n$, we see that the formula for $U$ holds true when applied to elements of $\{\phi_n\}_{n\in\N}$, and thus for all polynomials.

This theorem happens to be central to the results that follow. The first application of it that we will need is for a compact formula for the composition operator, whose expanded form was known to Pincherle and Amaldi <cit.>.

If $f$ is a power series,

\[
C_f = (e^{f(\x) - \x})^D.
\]

We apply the right-hand side on a monomial

\[
(e^{f(\x) - \x})^D x^n = \sum_{k=0}^\infty (f(\x) - \x)^k \frac{D^k}{k!} x^n = \sum_{k=0}^n \binom{n}{k} (f(x) - x)^k x^{n-k} = (f(x) - x + x)^n = C_f x^n.
\]

Along the same lines of operator identities, we give this famous extension of the shift operator.

For a power series $v$, if $h$ is a function such that $h'(x) = 1/v(x)$, then for all $t$

\[
e^{v(\x)Dt} = C_{h^{-1}(h(x)+t)},
\]

where $h^{-1}(h(x) + t)$ is a power series in $x$.

Let $f$ be a polynomial and $f(x) = F(h(x))$, then by the chain rule and the formula of <Ref>

\begin{align*}
e^{v(\x)Dt} f(x) &= e^{v(\x) h'(\x) D_h t} F(h) = E_h^t F(h) = F(h+t) \\
&= f(h^{-1}(h(x) + t)) = C_{h^{-1}(h(x)+t)} f(x).
\end{align*}

In <cit.>, a new kind of endomorphism $\LL$ on the set of operators was constructed. We give a characterization of a special instance of this operator <cit.> for $\L(\P)$.

There exists an operator $\LL : \L(\P) \to \L(\P)$ such that

* $\LL$ is linear.

* For all $U, V\in\L$, $\LL(UV) = \LL(V) \LL(U)$.

* $\LL(\x) = D$ and $\LL(D) = \x$.

In particular, $\LL^{-1} = \LL$, and for all power series $f$, $\LL(f(U)) = f(\LL(U))$.

This operator enables the transformation of operator identities into new forms that would otherwise be very difficult to prove. As an example and application of this function, we present the following corollary.

For $V$ a shift-invariant operator, if $h$ is a function such that $h'(x) = 1/\tilde V(x)$, then for all $t$

\[
e^{\x V t} = (e^\x)^{h^{-1}(h(D) + t) - D}.
\]

Starting from <Ref> and using the formula from <Ref>, we apply $\LL$ and use its properties

\begin{align*}
\LL(e^{v(\x)Dt}) &= \LL((e^{h^{-1}(h(\x) + t) - \x})^D), \\
e^{\LL(v(\x)Dt)} &= \sum_{k=0}^\infty \inv{k!} \LL((h^{-1}(h(\x) + t) - \x)^k D^k), \\
e^{\LL(D)\LL(v(\x))t} &= \sum_{k=0}^\infty \inv{k!} \LL(D^k) \LL((h^{-1}(h(\x) + t) - \x)^k), \\
e^{\x v(\LL(\x))t} &= \sum_{k=0}^\infty \inv{k!} \LL(D)^k (h^{-1}(h(\LL(\x)) + t) - \LL(\x))^k, \\
e^{\x v(D)t} &= (e^\x)^{h^{-1}(h(D) + t) - D},
\end{align*}

where we set $v = \tilde V$.

As an illustrative application, we have this general commutation identity, generalizing the formula $\phi x^n = \phi_n(x)$.

If $\phi \leadsto Q$ then for $n\in\N$

\[
\phi \x^n = (\x (Q')^{-1})^n \phi = \sum_{k=0}^n \x^k  B_{n, k}((\tilde Q^{-1})^{(1)}(Q), \ldots, (\tilde Q^{-1})^{(n-k+1)}(Q)) \phi.
\]

First, we use <Ref> $n$ times

\[
\phi \x^n =\x (Q')^{-1} \phi \x^{n-1} = \ldots = (\x (Q')^{-1})^n \phi.
\]

We can sum up the previous equation with a factor $t^n/n!$ to obtain the equivalent formula

\[
\phi e^{\x t} = e^{\x (Q')^{-1} t} \phi.
\]

<Ref> allows us to write

\begin{equation}\label{e:expQ'-1}
e^{\x (Q')^{-1}t} = (e^\x)^{\tilde Q^{-1}(Q + t) - D}.
\end{equation}

By the Taylor expansion

\[
\tilde Q^{-1}(Q + t) - D = \sum_{n=0}^\infty (\tilde Q^{-1})^{(n)}(Q) \frac{t^n}{n!} - D =  \sum_{n=1}^\infty (\tilde Q^{-1})^{(n)}(Q) \frac{t^n}{n!},
\]

\begin{align*}
(e^\x)^{\tilde Q^{-1}(Q + t) - D} &=
\sum_{k=0}^\infty \frac{\x^k}{k!} \bra{\sum_{n=1}^\infty (\tilde Q^{-1})^{(n)}(Q) \frac{t^n}{n!}}^k \\
&= \sum_{k=0}^\infty \x^k \sum_{n=k}^\infty B_{n, k}((\tilde Q^{-1})^{(1)}(Q), \ldots, (\tilde Q^{-1})^{(n-k+1)}(Q))  \frac{t^n}{n!} \\
&= \sum_{n=0}^\infty \frac{t^n}{n!} \sum_{k=0}^n \x^k  B_{n, k}((\tilde Q^{-1})^{(1)}(Q), \ldots, (\tilde Q^{-1})^{(n-k+1)}(Q)).
\end{align*}

Lastly, we will rapidly mention the Cauchy-Euler operator $\x D$. Its eigenvectors are famously the monomials as $\x D x^n = n x^n$. This is a special case of <Ref>.
Viskov and Srivastava noticed in [36] that, by linearity, we could extend this identity for a power series $f$ to

\begin{equation}\label{e:xD}
f(\x D) x^n = f(n) x^n.
\end{equation}

They mentioned that, in particular, if $f$ is the falling factorial, then $(\x D)_k x^n = (n)_k x^n$, which can be equated to the trivial equation $\x^k D^k x^n = (n)_k x^n$ to yield the well-known operator identity

\begin{equation}\label{e:(xD)^k}
(\x D)_k = \x^k D^k.
\end{equation}

§.§ A special class of umbral operators

After defining the coefficients in <Ref>, it seems natural to write umbral operators $\phi$ in the form

\begin{equation}\label{e:special}
\phi \x^n = \sum_{k=0}^n \coeff{n}{k}_\phi \x^k O(n, k) \phi,
\end{equation}

where $O(n,k)$ is shift-invariant and satisfy $O(n,k) 1 = 1$, because $\phi x^n = \phi_n(x) = \sum_{k=0}^n \coeff{n}{k}_\phi x^k$. We can identify with $O(n, k)$ with <Ref>,

\begin{equation}
\coeff{n}{k}_\phi O(n,k) = B_{n, k}((\tilde Q^{-1})^{(1)}(Q), \ldots, (\tilde Q^{-1})^{(n-k+1)}(Q)).
\end{equation}

For a particular class of umbral operators, it appears that $O$ takes the elegant form $O(n, k) = U^n V^k$. For example, with $\varphi$, one finds $U = 1$ and $V = E^{-1}$ due to the formula derived in <Ref>. In what follows, we will fully characterize operators of this form. Nevertheless, prior to the exposition, we require the characterization of functions satisfying a generalization of Cauchy's famous functional equation.

Let $f$ be a formal power series of the form

\[
f(x) = \sum_{k=1}^\infty \frac{c_k}{k!} x^k, \quad c_1 \neq 0.
\]

There exist formal power series $u, v$ such that for all $x, y$, $f$ satisfies the functional equation

\[
f(x+y) = f(x) + u(x) f(v(x) y) \tag{$*$}
\]

if and only if $f$ is of one of the following forms below. Furthermore, we have the expressions for $u$ and $v$

\[
\begin{array}{c|c|c}
f(x) & u(x) & v(x) \\
\hdoubleline
ax & g(x) & g(x)^{-1} \\
\hline
\rule{0pt}{10pt}
a(e^{bx} - 1) & e^{bx} & 1 \\
\hline
a\log(1+bx) & 1 & (1+bx)^{-1} \\
\hline
a((1+bx)^c - 1) & (1+bx)^c & (1+bx)^{-1}
\end{array}
\]

provided $a, b, c \neq 0$ and $g$ is any power series of order $0$.

Considering the form of the functional equation, it is not surprising to see that the solutions contain all power series that have a simple property that relates products and sums.

After differentiation with respect to $y$ once and twice of ($*$) at $0$

\[
\begin{cases}
f'(x) = u(x)v(x)c_1 \\
f''(x) = u(x)v(x)^2c_2.
\end{cases}
\]

If $c_2 = 0$, then $f''(x) = 0$ and with the additional fact that $f(0) = 0$, we get $f(x) = c_1 x$, for which if $u(x)v(x) = 1$, then ($*$) satisfied.
If $c_2 \neq 0$, then we can solve for $u$ and $v$

\[
\begin{cases}
u(x) = \bra{\dfrac{f'(x)}{c_1}}^2 \dfrac{c_2}{f''(x)} \\[11px]
v(x) = \dfrac{f''(x)}{c_2} \dfrac{c_1}{f'(x)}.
\end{cases}
\]

($*$) can be rearranged, and by replacing $y$ by $y f'(x)/c_1$, we get

\[
\frac{f''(x)}{c_2} \bra{f\bra{x + \frac{f'(x)}{c_1} y} - f(x)} = \bra{\frac{f'(x)}{c_1}}^2 f\bra{\frac{f''(x)}{c_2} y}.
\]

Differentiating with respect to $x$ at $0$ gives

\[
\frac{c_3}{c_2} f(y) + \bra{1+ \frac{c_2}{c_1}y} f'(y) - c_1 = 2 \frac{c_2}{c_1^2} f(y) + \frac{c_3}{c_2}y f'(y).
\]

Finally, we just need to solve the linear differential equation in $y$

\[
(1+ ay)f'(y) = b f(y) + c_1, \quad f(0) = 0,
\]

\[
a = \frac{c_2}{c_1} - \frac{c_3}{c_2} \quad\et\quad b = 2 \frac{c_2}{c_1^2} - \frac{c_3}{c_2}.
\]

There are four cases, but the (linear) case $a = b = 0$ was already covered.

* If $a = 0$ and $b \neq 0$, then $f(y) = c_1 \dfrac{e^{by}-1}{b}$.

* If $a \neq 0$ and $b = 0$, then $f(y) = c_1 \dfrac{\log(1+ay)}{a}$.

* If $a \neq 0$ and $b \neq 0$, then $f(y) = \dfrac{c_1}{b}((1+ax)^{b/a}-1)$.

All of these cases are equivalent to the ones in the lemma, and this concludes the proof.

If $\phi \leadsto Q$, then $\phi$ satisfies for some shift-invariant operators $U, V$, the commutation identity

\[
\phi \x^n = \sum_{k=0}^n \coeff{n}{k}_\phi \x^k U^k V^n \phi,
\]

if and only if $Q$ is of one of the following forms. In addition, we have the expressions for $U$ and $V$.

\[
\begin{array}{c|c|c}
Q & U & V \\
\hdoubleline
aD & A & A^{-1} \\
\hline
\rule{0pt}{10pt}
a (E^b-1) & 1 & E^{-b} \\
\hline
a \log(1+bD) & 1+bD & 1 \\
\hline
a((1+bD)^c-1) & 1+bD & (1+bD)^{-c}
\end{array}
\]

provided $a, b, c \neq 0$ and $A$ being any Appell operator.

It is worth mentioning that because of the form of the delta operators, it is not difficult to work out that the special class of umbral operators is invariant under inversion.

In the same way as in the proof of <Ref>, we can obtain

\[
\phi e^{\x t} = e^{\x (Q')^{-1} t} \phi = (e^\x)^{\tilde Q^{-1}(Q+t)-D} \phi. \tag{$*$}
\]

And we want to be able to write $\tilde Q^{-1}(Q + t) - D = U \tilde Q^{-1}(Vt)$. If we set $u \defeq \tilde U \circ \tilde Q^{-1}$ and $v \defeq \tilde V \circ \tilde Q^{-1}$, then by the isomorphism theorem (<Ref>) and some rearrangement, it can be written equivalently for all $x$ as

\[
\tilde Q^{-1}(x + t) = \tilde Q^{-1}(x) + u(x) \tilde Q^{-1}(v(x)t),
\]

hence $\tilde Q^{-1}$ follows the functional equation of <Ref>, hence for $a, b, c \neq 0$, $Q^{[-1]}$ is either $a(E^b-1)$, $a\log(1+bD)$ or $a((1+bD)^c - 1)$, and $U = u(Q)$, $V = v(Q)$, from which we can find the equivalent forms of the theorem. For these expressions for $Q$, we thus have, thanks to <Ref>

\begin{align*}
e^{\x (Q')^{-1} t} &= (e^\x)^{U\tilde Q^{-1}(Vt)} = \sum_{k=0}^\infty \frac{\x^k}{k!} (U \tilde Q^{-1}(Vt))^k \\
&= \sum_{k=0}^\infty \x^k U^k \g_\phi(k, Vt) = \sum_{k=0}^\infty \x^k U^k \sum_{n=k}^\infty \coeff{n}{k}_\phi \frac{(Vt)^n}{n!} \\
&= \sum_{n=0}^\infty \frac{t^n}{n!} \sum_{k=0}^n \coeff{n}{k}_\phi \x^k U^k V^n,
\end{align*}

and by identifying the coefficients in front of $t^n/n!$ in ($*$), we obtain the desired identity.

Thus, we have found all the umbral operators belonging to the special class when $O(n, k) = U^k V^n$. And as expected from <Ref>, it contains $\varphi$.

§.§ Explicit formulas for umbral operators

The topic of closed forms for umbral operators is challenging. Except for the identity, they are not shift-invariant, making them inapplicable to the Expansion theorem. Rota expressed the thought of solving “Pincherle differential equations” <cit.> umbral operators were satisfying, to find their expressions. However, due to all of the noncommutative issues, this idea was never developed and probably never will. The first formula was published by Garsia [13], although it can be directly derived from Corollary 7 of [30].

If $\phi \leadsto Q$,

\[
\phi^{-1} = \sum_{k=0}^\infty \frac{\x^k}{k!} \E Q^k.
\]

This result is in fact its own extension of the Taylor expansion in a different direction from the expansion theorem.

Expanding $E^y$ into powers of $Q$ via the expansion theorem (<Ref>), and applying the result to a polynomial $p(x)$ yields

\[
p(x+y) = \sum_{n=0}^\infty \frac{\phi_n(y)}{n!} Q^k p(x),
\]

which we use in this calculation

\[
\phi^{-1} p(x) = \phi^{-1}_x \E_y p(y+x) =  \phi^{-1}_x \E_y \sum_{n=0}^\infty\frac{\phi_n(x)}{n!} Q_y^k p(y) = \sum_{n=0}^\infty \frac{x^n}{n!} \E_y Q_y^k p(y) = \sum_{n=0}^\infty \frac{\x^n}{n!} \E Q^k p(x),
\]

to then notice the equality of the operators.

The proof suggests an obvious generalization, which we do not state, since the special case above is the simplest, and umbral operators are better manipulated in their simplest forms. This formula is, in fact, the definition of the inverse of a basic sequence used in <cit.> (referred to as conjugates).

This formula, while simple to use, has the drawback of including the evaluation operator $\E$ which is impractical due to its limited set of identities. Di Bucchianico and Loeb, using the Kurbanov-Maksimov expansion (<Ref>), derived a more elegant formula expressed solely in terms of $\x$ and $D$. Furthermore, it can be reformulated in a more compact and aesthetically pleasing manner using our notation.

If $\phi \leadsto Q$,

\[
\phi = (e^\x)^{Q^{[-1]} - D}.
\]

After using <Ref>

\[
\frac{\G_\phi(x, t)}{\G_1(x, t)} = e^{x(\tilde Q^{-1}(t) - t)} = \sum_{k=0}^\infty \inv{k!} x^k (\tilde Q^{-1}(t) - t)^k,
\]

hence by Kurbanov-Maksimov expansion (<Ref>).

\[
\phi = \sum_{k=0}^\infty \inv{k!} \x^k (\tilde Q^{-1}(D) - D)^k = (e^\x)^{Q^{[-1]} - D}.
\]

Most applications of this theorem will not lead to useful closed forms. For example, $\varphi = (e^\x)^{\log(1+D) - D}$. However, if we take the stretched derivative $Q = D/\lambda$, then $Q^{[-1]} = \lambda D$ and by <Ref>

\[
\str\lambda = \sum_{k=0}^\infty \frac{\x^k}{k!} (\lambda D - D)^k = \sum_{k=0}^\infty \frac{\x^k D^k}{k!} (\lambda - 1)^k = \sum_{k=0}^\infty \frac{(\x D)_k}{k!} (\lambda - 1)^k = \lambda^{\x D}.
\]

This shows that the equation $\lambda^{\x D} x^n = \lambda^n x^n$ is completely reminiscent of <ref>.

The closed forms for basic sequences (<Ref>) have existed for 80 years and we can express them solely in terms of operators. Though one cannot just identify them as such (because the umbral operator can't depend on $n$), we are able to use the simple fact of <ref> to do something analogous.

If $\phi \leadsto Q$,

\[
\phi = Q'(D/Q)^{\x D+1} = \x (D/Q)^{\x D+1} \x^{-1}.
\]

It is important to remember that we only have

\begin{equation}
(D/Q)^{xD+1} = \frac{D}{Q} (D/Q)^{xD},
\end{equation}

and generally not $(D/Q)^{xD+1} = (D/Q)^{xD} \frac{D}{Q}$. This is a consequence of the convention that we chose in <ref>. This convention is justified here because the alternative convention would lead to the less elegant formula “$\phi = Q' (e^{\x D+1})^{\log(D/Q)}$”.

By the definition of the exponentiation and <Ref>,

\begin{align*}
&Q' (D/Q)^{\x D} x^n = Q' \sum_{k=0}^\infty (D/Q - 1)^k \binom{\x D+1}{k} x^n \\
={}& Q' \sum_{k=0}^\infty (D/Q - 1)^k \frac{(n+1)_k}{k!} x^n = Q' (D/Q)^{n+1} x^n \\
={}& \phi_n(x) = \phi x^n.
\end{align*}

The second formula has an identical proof.

Deriving the equality $\str\lambda = \lambda^{\x D}$ is even more immediate with this formula. But apart from that, this formula is quite inconsequential.

Now we want to investigate the formula for umbral operators in the simplest form possible, i.e.

\[
\phi = e^{\x V} = 1 + \x V + \inv2 (\x V)^2 + \ldots,
\]

$V$ being shift-invariant. $V$ also must be a strictly degree-reducing operator, otherwise $\phi$ would map polynomials to power series. For example, with $V = D$ we have $e^{\x D} = \str{e}$.

First, given the operator $e^{\x V}$, if we define $h(t) = \int^t 1/\tilde V$, we can use <Ref> and identify the result with <Ref> to find that the power series resulting from the simplification of $h^{-1}(h(t) - 1)$ is the indicator of the delta operator for $e^{\x V}$. The constant term of $h$ does not affect the formula for the delta operator and should be chosen to prevent divergence.

However, finding $V$ in terms of the delta operator proves to be a much more difficult task, for which we require the following elegant lemma.

If $\phi\leadsto Q$ then

\[
\LL(C_{\tilde Q^{-1}}) = \phi.
\]

This result raises the question of what interpretation we can give to the action of $\LL$.

This result follows from <Ref> and <Ref>.

Let $\phi$ be basic for $Q$. Then

\[
\phi = e^{-\x Q_*},
\]

where $Q_* \defeq \itlog(\tilde Q)(D)$.

We simply combine the previous lemma with the formula of <Ref> (d).

If $Q = D/\lambda$, then of course $Q^{[s]} = D/\lambda^s$ and

\[
Q_* = \left.\odv{Q^{[s]}}{s}\right|_{s=0} = - \log(\lambda) D,
\]

therefore, with <Ref>, we find once again that $\str\lambda = \lambda^{\x D}$.

In light of this theorem, fractional exponents of umbral operators can be defined as $\phi^s \defeq e^{-s\x Q_*}$, maintaining their status as umbral operators. Consequently, this enables us to define its unique delta operator $Q^{[s]}$. These notations are consistent with the previous notation when $s$ is an integer and the indicator of $Q^{[s]}$ clearly coincides with the fractional iterations of $\tilde Q$, in the sense of Écalle. Furthermore, the coefficients of $\phi^s$ are identified from <Ref> with <Ref> to be for $s\in\K$

\begin{equation}
\coeff{n}{k}_{\phi^s} = \sum_{p=0}^{n-k} \binom{s}{p} \coeff{n}{k}_{(\phi - 1)^p} =  \sum_{p=0}^{n-k} \binom{s}{p} \binom{n-k-s}{n-k-p} \coeff{n}{k}_{\phi^p}.
\end{equation}

$(Q^{[s]})_{s\in\K}$ also forms a one-parameter group under $\diamond$ with infinitesimal generator $- \x Q_*$. Thus, what precedes answers completely Rota's third problem <cit.>.

§.§ Sigma operators

Although it is a simple consequence of <Ref> (a) and <Ref> (a) that delta operators are not invertible, it seems entirely natural to define some kind of antiderivative analog to the derivative but for other delta operators. This idea dates back to Steffensen [35] where he defined the antiderivative $\bar D$ such that $D \bar D = 1$ and $(\bar D f)(0) = 0$ for any polynomial $f$, and extended the definition to delta operators via $\bar Q = \bar D (D/Q)$. Steffensen realized that $D$ and $\bar D$ did not commute since $\bar D D f(x) = f(x) - f(0)$. It is upon this fact that we will define pseudoinverses of delta operators. We shall call them sigma operators.

For any delta operator $Q$, there exists a unique linear operator $Q^{-1}$ such that

\begin{equation}
Q Q^{-1} = 1 \quad \et \quad Q^{-1} Q = 1 - \E.
\end{equation}

Moreover, if $\{s_n\}_{n\in\N}$ is a Sheffer set for $Q$,

\begin{equation}\label{e:sigmasheffer}
Q^{-1} s_n(x) = \frac{s_{n+1}(x) - s_n(0)}{n+1}.
\end{equation}

We call $Q^{-1}$ the sigma operator associated with $Q$.

Let $\{p_n\}_{n\in\N}$ be the basic set for $Q$. Define $Q^{-1}$ by $Q^{-1} p_n(x) = \inv{n+1}p_{n+1}(x)$ and extended by linearity on all power series. So by the binomial theorem (<Ref>)

\begin{align*}
Q^{-1} s_n(x)
&= Q^{-1} \sum_{k=0}^n \binom{n}{k} s_{n-k}(0) p_k(x) = \sum_{j=1}^{n+1} \binom{n}{j-1} s_{n-j+1}(0) \frac{p_j(x)}{j} \\
&= \inv{n+1} \sum_{j=1}^{n+1} \binom{n+1}{j} s_{n+1-j}(0) p_j(x) = \frac{s_{n+1}(x) - s_{n+1}(0)}{n+1},
\end{align*}

because $p_0(x) = 1$. Now we verify that it follows the said properties

\begin{align*}
&QQ^{-1} f(x) = Q \sum_{k=0}^\infty \frac{a_k}{k+1} p_{k+1}(x) \\
&= \sum_{k=0}^\infty a_k p_k(x) = f(x)
\end{align*}

\begin{align*}
&Q^{-1} Q f(x) = Q^{-1} \sum_{k=1}^\infty a_k k p_{k-1}(x) \\
&= \sum_{k=1}^\infty a_k p_k(x) = f(x) - a_0 = (1-\E)f(x)
\end{align*}

Thus, $Q^{-1}$ exists, and to show the unicity, suppose $I_Q$ is another sigma operator, then, in particular, $Q^{-1} Q = I_Q Q$. Because $Q$ is surjective, $Q^{-1} = I_Q$.

If $\phi \leadsto Q$, for positive integers $n$ and $p$

\[
(n+1) \cdots (n+p) Q^{-p} \phi_n(x) = \phi_{n+p}(x),
\]

where $Q^{-p} \defeq (Q^{-1})^p$. In particular, $\phi_n(x) = n! Q^{-n} 1$.

Sigma operators can be easily extended by defining $Q^{-1}_{(a)} \defeq E^{-a} Q^{-1} E^a$. They are characterized by

\begin{equation}
Q Q^{-1}_{(a)} = 1, \quad \and \quad Q^{-1}_{(a)} Q = 1 - \Ev a.
\end{equation}

This extension allows for a broader application of sigma operators.

An unsurprising exact formula for the integral operator $\I \defeq D^{-1}$ applied to functions is

\[
\Int a f:x\longmapsto \int_a^x f(t) \odif t,
\]

which holds by the unicity. $\Int a D = 1 - \Ev a$ is the operational translation of the fundamental theorem of calculus.

For the sum operator $\Sigma \defeq \Delta^{-1}$, we have the natural formula to compute it, but it only works when $a$ and $x$ are integers

\[
\Sum a f : x \longmapsto \sum_{k=a}^{x-1} f(k),
\]

and $\Sum a \Delta = 1 -\Ev a$ is simply a rewording of the telescopic sum. We will discuss later how $\Sum a$ actually constitutes a generalization of the summation operation to fractional indices.

Since sigma operators act as pseudo-inverses, it is logical to assume they are connected to general inverses and operator division. This assumption is indeed correct.

For $Q, R$ two delta operators, these identities hold

* $\Ev{a} Q^{-1}_{(a)} = 0$.

* For an Appell operator $A$, $(AQ)^{-1}_{(a)} = Q^{-1}_{(a)} A^{-1}$.

* $Q^{-1}_{(a)} = R^{-1}_{(a)} \dfrac{R}{Q}$.

* $Q/R = Q R^{-1}_{(a)} = Q R^{-1}$.

First, by definition

\[
Q^{-1}_{(a)} = Q^{-1}_{(a)} (Q Q^{-1}_{(a)}) = (1 - \Ev{a}) Q^{-1}_{(a)} = Q^{-1}_{(a)} - \Ev{a} Q^{-1}_{(a)},
\]

from which we deduce (a). Furthermore, (a) implies (b) when we use that $AQ (AQ)^{-1}_{(a)} = 1$ and apply $Q^{-1}_{(a)} A^{-1}$ on the left

\[
Q^{-1}_{(a)} A^{-1} = Q^{-1}_{(a)} Q (AQ)^{-1}_{(a)} = (1 - \Ev{a}) (AQ)^{-1}_{(a)} = (AQ)^{-1}_{(a)}.
\]

Setting $A = R/Q$ (which is an Appell operator by <Ref>) we obtain (c) with <ref>. (d) follows by applying $R$ to the left.

With <Ref> (d), we can obtain an integral formula for the inverse of the Bernoulli operator $\Bern^{-1}$ since by <Ref>

\[
\Bern^{-1} = \Delta/D = \Delta \Int a.
\]

\[
\Bern^{-1} f(x) = \int_a^{x+1} f(t) \odif t -  \int_a^x f(t) \odif t =  \int_x^{x+1} f(t) \odif t.
\]

[Faulhaber's formula]
Bernoulli numbers, defined in <Ref>, notably find utility in expressing the sum of the first $n$-th powers. To find the formula, we apply identity (c) of <Ref>, that is, $\Sigma = \I \Bern$, to the monomials

\begin{equation}
\sum_{k=0}^{x-1} k^n = \int_0^x \Bern_n(t) \odif t = \frac{\Bern_{n+1}(x) - \Bern_{n+1}(0)}{n+1} =  \frac{\Bern_{n+1}(x) - \Bern_{n+1}}{n+1},
\end{equation}

because of <ref>. By <Ref> we also have the following

\begin{equation}
\sum_{k=0}^{x-1} k^n = \int_0^x \Bern_n(t) \odif t = \int_0^x \sum_{k=0}^n \binom{n}{k} \Bern_k(0)  t^{n-k} \odif t = \sum_{k=0}^n \binom{n}{k} \Bern_k \frac{x^{n+1-k}}{n+1-k}.
\end{equation}

Both formulas can be transformed into a better closed form, also known as Faulhaber's formula

\begin{equation}
\sum_{k=0}^{x-1} k^n = \inv{n+1} \sum_{k=0}^n \binom{n+1}{k} \Bern_k x^{n+1-k}.
\end{equation}

For two delta operators $Q$ and $R$, if you can write $Q/R = \sum a_k Q^k$, then

\[
R^{-1}_{(a)} - a_0 Q^{-1}_{(a)} = \sum_{k=1}^\infty a_k (1-\Ev{a}) Q^{k-1}.
\]

We simply apply <Ref> (c) and expand

\[
R^{-1}_{(a)} = Q^{-1}_{(a)} \frac{Q}{R} = Q^{-1}_{(a)} \sum_{k=0}^\infty a_k Q^k = a_0 Q^{-1}_{(a)} + \sum_{k=1}^\infty a_k (1-\Ev{a}) Q^{k-1}.
\]

The Bernoulli operator can be expanded with Bernoulli numbers :

\begin{equation}
\Bern = D/\Delta = \frac{D}{e^D - 1} = \sum_{k=0}^\infty \frac{\Bern_k}{k!} D^k,
\end{equation}

applying <Ref>, we obtain the operational form of the classical Euler-Maclaurin formula

\begin{equation}
\Sum a - \Int a = \sum_{k=1}^\infty \frac{\Bern_k}{k!} (1-\Ev a) D^{k-1}.
\end{equation}

If $\phi \leadsto Q$. Let $s = \dfrac{Q}{R}\phi$ with $R$ being another delta operator. then

\[
R^{-1}_{(a)} = \sum_{n=1}^\infty \frac{s_n(0)-s_n(a-\x)}{n!} Q^{n-1}.
\]

The formula containing the operator $\x$ supports the fact that sigma operators are not shift-invariant.

By <Ref> (c)

\begin{align*}
R^{-1}_{(a)} G_\phi(x, t) &= Q^{-1}_{(a)} \frac{Q}{R} e^{x \tilde Q^{-1}(t)} = Q^{-1}_{(a)} \G_s(x,t) = \sum_{n=0}^\infty \frac{Q^{-1}_{(a)} s_n(x)}{n!}t^n \\
&= \sum_{n=0}^\infty \frac{s_{n+1}(x) - s_{n+1}(a)}{(n+1)!}t^n = \frac{\G_s(x, t) - \G_s(a, t)}{t},
\end{align*}

because $s$ is a Sheffer set for $Q$. Now to apply the Kurbanov-Maksimov expansion (<Ref>) we perform this calculation, keeping in mind <Ref>

\[
\frac{R^{-1}_{(a)} G_\phi(x, t)}{G_\phi(x, t)} = \frac{\G_s(0, t) - \G_s(a-x, t)}{t} = \sum_{n=1}^\infty \frac{s_n(0) - s_n(a - x)}{n!} t^{n-1},
\]

whence the result.

It is more convenient to write $Q^{-1}_{(a)}$ in terms of $Q$ as $\phi_n(0) = 0$ for $n > 0$.

If $\phi \leadsto Q$,

\[
Q^{-1}_{(a)} = -\sum_{n=1}^\infty \frac{\phi_n(a-\x)}{n!} Q^{n-1}.
\]

For the integration operator, we obtain

\begin{equation}
\Int a = -\sum_{k=1}^\infty \frac{(a-\x)^k}{k!} D^{k-1},
\end{equation}

which was featured in [6] (in the case $a=0$), contrary to the very remarkable formula for the discrete analogue

\begin{equation}
\Sum a = -\sum_{k=1}^\infty \binom{a-\x}{k} \Delta^{k-1} = \sum_{k=0}^\infty \binom{\x - a + k}{k+1} (-\Delta)^k.
\end{equation}

The surprising aspect is that recalling the definition of $\Sum a$ in <Ref>, we understand it sums the values of its argument between two integer bounds. However, this formula allows sums to be extended to any bounds, including complex ones. It does not carry the same intuition that we have with integer bounds and should not be confused with summation over some set $\sum_{k\in S}$. Instead, it represents a different kind of operation. For example, using <Ref> (c) and <Ref>

\begin{equation}
\Sigma e^{xt} = \I \Bern e^{xt} = \frac{t}{e^t - 1} \I e^{xt} = \frac{t}{e^t - 1} \frac{e^{xt}-1}{t} = \frac{e^{xt}-1}{e^t-1},
\end{equation}

which is the fractional generalization of the geometric sum formula

\[
\sum_{k=0}^{x-1} e^{kt} = \sum_{k=0}^{x-1} (e^t)^k = \frac{e^{xt}-1}{e^t-1},
\]

where $x$ is a positive integer and $t$ is nonzero. Müller and Schleicher have previously established a successful theory of fractional sums in [26]. Their theory bears similarities with the fractional sums that arise as a consequence of the Ramanujan summation technique <cit.>. Therefore, it would be intriguing to explore the connections, and possibly establish equivalence, between $\Sigma$ and the previously developed fractional sums.

§ EXAMPLES

To thoroughly demonstrate the effectiveness of the previously developed tools, we will explore some of the most important basic sets, including Abel, Touchard, and Laguerre polynomials, after revisiting the more recent example <cit.>. Specifically, through quick computations, we will derive both old and new identities, some of which previously lacked umbral proofs. Furthermore, a new generalization of Laguerre polynomials will be introduced, which naturally arises from <Ref> and is referred to as degenerate.

§.§ A summarizing example featuring Catalan numbers

The logistic derivative defined by $\Lambda \defeq D(1-D)$ can be easily checked to be a delta operator. We define the Catalan operator by $\Ca\leadsto\Lambda$. The coefficients of Catalan polynomials can be quickly computed with the Steffensen formula (<Ref>)

\begin{align}
\Ca_n(x)
&= \x \bra{\frac{D}{\Lambda}}^n x^{n-1} = \x (1-D)^{-n} x^{n-1} = \x \sum_{k=0}^\infty \binom{-n}{k} (-D)^k x^{n-1} \nonumber\\
&= \sum_{k=0}^n \binom{n+k-1}{n-1} \frac{(n-1)!}{(n-1-k)!} x^{n-k} = \sum_{k=0}^n \binom{2n-k-1}{n-1} \frac{(n-1)!}{(k-1)!} x^k.
\end{align}

Because we can write

\begin{equation}\label{e:Lambda}
\Lambda = \frac{1 - (1-2D)^2}{4},
\end{equation}

the Catalan operator is part of the special class of umbral operators, thus, by <Ref>, it is immediate that.

\begin{equation}
\Ca \x^n = \sum_{k=0}^n \binom{2n-k-1}{n-1} \frac{(n-1)!}{(k-1)!} \x^k (1-2D)^{k-2n} \Ca.
\end{equation}

Inversion of <ref> yields

\begin{equation}\label{e:Lambda-1}
\Lambda^{[-1]} = \dfrac{1-\sqrt{1-4D}}{2} = \sum_{n=1}^\infty C_{n-1} D^n,
\end{equation}

where $C_n$ is the $n$-th the Catalan number. Without <Ref>, it would be much more difficult to find the formula for the following power series

\begin{equation}
\bra{\dfrac{1-\sqrt{1-4D}}{2}}^k = \sum_{n=k}^\infty \frac{k}{n} \binom{2n-k-1}{n-1} D^n,
\end{equation}

and constitutes yet another way to recover the famous formula for the Catalan numbers (with $k=1$)

\begin{equation}
C_n = \inv{n+1} \binom{2n}{n}.
\end{equation}

Moreover, we can also infer the evaluation of Bell polynomials at Catalan numbers

\begin{equation}
B_{n,k}(C_0, \ldots, C_{n-k}) = \frac{(n-1)!}{(k-1)!} \binom{2n-k-1}{n-1}.
\end{equation}

Finding the inverse $\Ca^{-1} \leadsto \Lambda^{[-1]}$ is made easier by <Ref>. Indeed

\begin{align*}
&\bra{\frac{\Lambda}{D}}^k x^n
= (1-D)^k x^n = \sum_{p=0}^k \binom{k}{p} (-D)^p x^n \\
={}& \sum_{p=0}^k \binom{k}{p} (-1)^p \frac{n!}{(n-p)!} x^{n-p} = \sum_{p=n-k}^n \binom{k}{n-p} \frac{n!}{p!} (-1)^{n-p} x^p,
\end{align*}

from which we recover the coefficient at index $p=k$

\begin{equation}
\coeff{n}{k}_{\Ca^{-1}} =
\begin{cases}
0 \com{if} k < \ceil{\dfrac{n}{2}} \\[9pt]
\displaystyle\binom{k}{n-k} \frac{n!}{k!} (-1)^{n-k} \com{if} k \geq \ceil{\frac{n}{2}},
\end{cases}
\end{equation}

keeping in mind that the binomial coefficient vanishes when $k \geq n-k$, or equivalently when $k \geq \ceil{n/2}$. The equality can be rewritten as

\begin{equation}
\Ca^{-1}_n(x) = \sum_{k=\ceil{n/2}}^n \binom{k}{n-k} \frac{n!}{k!} (-1)^{n-k} x^k.
\end{equation}

$\Ca^{-1}$ is also part of the special class, hence, looking at <ref>, we obtain more generally

\begin{equation}
\Ca^{-1} \x^n = \sum_{k=\ceil{n/2}}^n \binom{k}{n-k} \frac{n!}{k!} (-1)^{n-k} \x^k (1-4D)^{k-n/2} \Ca^{-1}.
\end{equation}

The sigma operator for $\Lambda$ can be expressed with integration thanks to <Ref> (c)

\begin{equation}
\Lambda^{-1} = D^{-1} \frac{D}{\Lambda} = \I \inv{1-D},
\end{equation}

and when applied to polynomials, this has a double integral representation

\begin{equation}
\Lambda^{-1} f(x) = \int_0^x \int_0^\infty e^{-t} f(z + t) \odif t \odif z,
\end{equation}

because of the known formula

\begin{equation}
\inv{1-D} = \int_0^\infty e^{-t} E^t \odif t.
\end{equation}

§.§ Abel polynomials and idempotent numbers

The Abel operator is defined for some fixed constant $a$ by $A \leadsto \D \defeq D E^a$. Once again, the formula of the shifted derivative suggests the use of the Steffensen formula

\begin{align}
A_n(x) &= \x (D/\D)^n x^{n-1} = \x E^{-an} x^{n-1} \nonumber\\
&= x(x-an)^{n-1}.
\end{align}

The Abel polynomials are basic; thus, they are also of binomial type (<Ref>) and Abel's identity is immediate

\begin{equation}
(x+y)(x+y-an)^{n-1} = \sum_{k=0}^n \binom{n}{k} x(x-ak)^{k-1} y(y-a(n-k))^{n-k-1}.
\end{equation}

By <Ref>, we can recover the closed form for the Abel umbral operator

\begin{equation}
A = \D' E^{-a} (E^{-a})^{\x D} = (1+aD) (E^{-a})^{\x D}.
\end{equation}

From this equation, we define the smooth Abel operator

\begin{equation}
\A \defeq (E^{-a})^{\x D} = (1+aD)^{-1} A.
\end{equation}

<Ref> assures us that $\A$ is a Sheffer operator for $\D$. It is given by

\begin{equation}
\A_n(x) = (E^{-a})^{\x D} x^n = (x-an)^n.
\end{equation}

Thanks to <Ref>, we can express the “smoother” Abel identity

\begin{equation}
(x+y-an)^n = \sum_{k=0}^n \binom{n}{k} x(x-ak)^{k-1} (y-a(n-k))^{n-k}.
\end{equation}

The sigma operator can also be expressed with integration after using <Ref> (c)

\begin{equation}
\D^{-1} = \I E^{-a}.
\end{equation}

The Niederhausen transform (<Ref>) allows a very quick computation of the inverse of $A$. Indeed, $\binom{n}{k}\str{a}_{n-k}(k) = \binom{n}{k}(ak)^{n-k}$ is the coefficients of a basic set, whose inverse is basic for $De^{aD} = \D$. So we precisely obtain

\begin{equation}
A^{-1}_n(x) = \sum_{k=0}^n \binom{n}{k}(ak)^{n-k} x^k,
\end{equation}

\begin{equation}
\D^{[-1]} = \sum_{n=1}^\infty \frac{(-an)^{n-1}}{n!} D^n.
\end{equation}

For $a=1$, the coefficients $\binom{n}{k}k^{n-k}$ that appear are known as idempotent numbers, and the power series for $\D^{[-1]}$ corresponds to the main branch of the Lambert W function.

§.§ Divided difference, rising factorials and Touchard polynomials

The divided difference operator is $\Delta_h = (E^h - 1)/h$. It generalizes $\Delta = \Delta_1$. We can extend the definition to $h=0$ by $\Delta_0 \defeq \lim_{h\to0} \Delta_h$, which, by the definition of the derivative, equals $D$. We can reduce the study of its basic sets by noticing that $\Delta_h = (D/h) \diamond \Delta \diamond (hD)$. Thus, by <Ref>, we deduce this formula for $\varphi_h \leadsto \Delta_h$

\begin{equation}\label{e:phih}
\varphi_h = \str{1/h} \varphi \str h.
\end{equation}

\begin{align}
\varphi_{h, n}(x) &\defeq (\varphi_h)_n(x) = \str{1/h} \varphi \str h x^n \nonumber\\
&= h^n (x/h)_n = x(x - h) \cdots (x - (n-1)h).
\end{align}

$\varphi_h$ is part of the special class of umbral operators of <Ref>, and we can work out the identity

\begin{equation}
\varphi_h \x^n = h^n (\x/h)_n E^{-hn} \varphi_h.
\end{equation}

This can also be proved by the repeated application of <Ref>.

The other important instance of $\varphi_h$ is the case $h = -1$. Indeed, the backward difference $\nabla \defeq \Delta_{-1} = 1 - E^{-1}$ is the delta operator for the rising factorials $\{\abra{x}_n\}_{n\in\N}$. If we denote the umbral operator with $\rho \defeq \varphi_{-1}$, then $\rho_n(x) = \abra{x}_n \defeq x (x+1) \cdots (x + n-1)$, or more generally

\begin{equation}
\rho \x^n = \abra \x_n E^n \rho.
\end{equation}

Much like we had this formula for the unsigned Stirling number of the second kind, $\Stir{n}{k} = \coeff{n}{k}_{\varphi^{-1}}$, we also have the analogous formula for the first kind $\coeff{n}{k} = \coeff{n}{k}_\rho$.

§.§.§ Bernoulli polynomials of the second kind

A known Sheffer set for $\Delta$ is the Bernoulli polynomials of the second kind. They can be defined with

\begin{equation}
\psi \defeq \Bern^{-1} \varphi.
\end{equation}

They are a Sheffer set for $\Delta$ based on <Ref>. We can use <Ref> to obtain a closed form formula

\begin{equation}
\psi_n(x) = \int_x^{x+1} (t)_n \odif t.
\end{equation}

Second kind Bernoulli polynomials satisfy this interesting commutation property

\begin{equation}
\varphi D = D \psi.
\end{equation}

§.§.§ Touchard polynomials

The Touchard polynomials (or exponential polynomials) $\{T_n\}_{n\in\N}$ are defined as

\begin{equation}
T_n(x) \defeq \varphi^{-1} x^n = \sum_{k=0}^n \Stir{n}{k} x^k,
\end{equation}

Therefore, $T = \varphi^{-1}$. Because $\Delta = e^D - 1$, from <Ref>, we deduce that the Touchard polynomials are a basic set of polynomials for the delta operator $\L \defeq \Delta^{[-1]} = \log(1+D)$.

Defining Bell numbers as follows

\begin{equation}
B_n \defeq T_n(1) = \sum_{k=0}^n \Stir{n}{k},
\end{equation}

we are able to deduce many of its properties. One notable result related to the higher-order exponentiation of $\varphi$, valid for $n > 0$, is

\begin{equation}\label{e:bellphi}
\coeff{n}{1}_{\varphi^{-2}} = B_n.
\end{equation}

It follows from <Ref>

\[
\coeff{n}{1}_{\varphi^{-2}} = \sum_{j=1}^n \coeff{n}{j}_{\varphi^{-1}} \coeff{j}{1}_{\varphi^{-1}} = \sum_{j=1}^n \Stir{n}{j} \Stir{j}{1},
\]

and the fact equations $\Stir{j}{1} = 1$ and $\Stir{n}{0} = \delta_n$. Nevertheless, the subsequent

\begin{equation}
\coeff{n}{k}_{\varphi^{-2}} = B_{n,k}(B_1, \ldots, B_{n-k+1}),
\end{equation}

following from <ref>, do not serve as a bivariate generalization of Bell numbers, since <ref> does not include the trivial case $B_0 = 1$. The formula is merely an incidental result rooted in the fact that exponentiation is a key function in umbral calculus.

The recurrence relation falls off <Ref> and <Ref>. Indeed, $(\L^{[-1]})' = E$, so

\begin{equation}
T_{n+1}(x) = \x \varphi^{-1} E x^n = x \sum_{k=0}^n \binom{n}{k} T_k(x).
\end{equation}

This operationally proves the equivalent umbral identity of <cit.>. By evaluating at $x = 1$, we obtain the well-known recurrence

\begin{equation}
B_{n+1} = \sum_{k=0}^n \binom{n}{k} B_k.
\end{equation}

The formula can be further generalized by firstly generalizing the operator formula, similarly to <Ref>.

\[
\varphi^{-1} \x^n = \sum_{k=0}^n \Stir{n}{k} \x^k \varphi^{-1} E^k.
\]

Use <Ref> in

\[
\sum_{k=0}^n \coeff{n}{k}_\varphi \varphi^{-1} \x^k = \varphi^{-1} (\x)_n  = \x^n \varphi^{-1} E^n,
\]

to recover the desired result after applying the first inversion formula (<Ref>).

Thus, $\varphi^{-1}$ is certainly part of the special class, so the identity is alternatively proved with <Ref> and the identity $(1+D) \varphi^{-1} = \varphi^{-1} E$.

This proposition straightforwardly implies Spivey's identity [34] and its generalization [14].

\[
T_{n+m}(x) = \sum_{k=0}^n \Stir{n}{k} x^k \sum_{j=0}^m \binom{m}{j} k^{m-j} T_j(x).
\]

Applying <Ref> to $x^m$

\[
T_{n+m}(x) = \sum_{k=0}^n \Stir{n}{k} \x^k \varphi^{-1} (x + k)^m = \sum_{k=0}^n \Stir{n}{k} x^k \sum_{j = 0}^m \binom{m}{j} k^{m-j} T_j(x).
\]

At $x=1$, we recover Spivey's original identity on Bell numbers

\begin{equation}
B_{n+m} = \sum_{k=0}^n \Stir{n}{k} \sum_{j=0}^m \binom{m}{j} k^{m-j} B_j.
\end{equation}

§.§.§ Dobiński's formula

Recall the identity of <ref> $\x^n D^n = (\x D)_n = \sum_{k=0}^n \coeff{n}{k}_\varphi (\x D)^k$. Therefore, by the first inversion formula (<Ref>)

\begin{equation}\label{e:xD Stir}
(\x D)^n = \sum_{k=0}^n \Stir{n}{k} \x^k D^k.
\end{equation}

\[
T_n(x) = e^{-x} \sum_{k=0}^\infty \frac{k^n x^k}{k!}.
\]

On the on hand

\[
(\x D)^n e^x = \sum_{k=0}^\infty (\x D)^n \frac{x^k}{k!} = \sum_{k=0}^\infty \frac{k^n x^k}{k!},
\]

and on the other hand, we get by <ref>

\[
(\x D)^n e^x = \sum_{k=0}^n \Stir{n}{k} \x^k D^k e^x = T_n(x) e^x,
\]

whence the result.

Looking at Dobiński's formula, it is clear that if we define the generating function linear operator by

\begin{equation}
\Gat f(x) \defeq \sum_{k=0}^\infty \frac{f(k)}{k!} x^k,
\end{equation}

then this identity follows.

\[
\varphi^{-1} = e^{-\x} \Gat.
\]

A first consequence of this is a surprising identity for the integer evaluation of $\varphi$.

If $f$ is a polynomial
\[
\varphi (f)(k) = \E D^k e^x f(x).
\]

The formula is obtained by rewriting the previous corollary (<Ref>) as $\Gat \varphi = e^\x$ and applying it to $f$

\[
\sum_{k=0}^\infty \frac{\varphi(f)(k)}{k!} x^k = e^x f(x),
\]

from which we identify the coefficients.

Furthermore, the Dobiński identity (<Ref>) allows us to find an elegant integral formula for $\rho$.

for a polynomial $f$ and for $x > -\deg f$

\[
\rho f(x) = \inv{\Gamma(x)} \int_0^\infty t^{x-1} e^{-t} f(t) \odif t.
\]

Using <ref> with $h = -1$ and <Ref>, we see that

\[
\sym\Gat\sym\rho = \sym\Gat\varphi\sym = \sym e^\x \sym = e^{-\x}
\]

and applying this to $f$ yields

\[
\sum_{k=0}^\infty \frac{\rho(f)(-k)}{k!} (-x)^k = e^{-x} f(x).
\]

Now we can apply Ramanujan's Master theorem [1] and obtain

\[
\rho(f)(x) \Gamma(x) = \int_0^\infty t^{x-1} e^{-t} f(t) \odif t.
\]

The condition of convergence on $x$ is based on the convergence domain of the integral representation of the $\Gamma$ function (which is for $x > 0$).

§.§ Laguerre polynomials and its new generalization

§.§.§ Degenerate associated Laguerre polynomials

If we consider the simplest families of umbral operators in the Écalle form, that is, $L_p \defeq e^{-\x D^{p+1}}$ for $p$ a positive integer, then by <Ref>, we can show that they are basic for $\Psi_p \defeq D/\sqrt[p]{1-pD^p}$, when $p > 0$ and $\Psi_0 \defeq e D$ (where $e$ is Euler's number) when $p = 0$. We denote the basic set by $\{L_{p, n}\}_{n\in\N}$ (with $L_{p, n}(x) \defeq (L_p)_n(x)$) and introduce the cross-sequence $\{\Lag\alpha_{p, n}\}_{n\in\N}$ defined by $\Lag\alpha_p \defeq (1-pD^p)^\alpha L_p$ and call them degenerate associated Laguerre polynomials as they generalize associated (or generalized) Laguerre polynomials, which we can retrieve with the case $p=1$ (see next section). Immediately from <Ref>, we recover

\begin{equation}
\Lag{\alpha + \beta}_{p,n}(x+y) = \sum_{k=0}^n \binom{n}{k} \Lag\alpha_{p,k}(x) \Lag\beta_{p,n-k}(y).
\end{equation}

In the following, we first compute $\Lag\alpha_{p,n}(x)$ for $p \geq 1$ (as the case $p=0$ is a completely known special case) and then deduce $L_p = \Lag{0}_p$. $\Psi_p' = 1/(1-pD^p)^{1/p+1}$, thus, with the transfer formula (<Ref>) we have $L_{p,n}(x) = \Psi_p' (\sqrt[p]{1-pD^p})^{n+1} x^n = (1-pD^p)^{n/p - 1} x^n$. Therefore

\begin{align}
\Lag\alpha_{p,n}(x) &= (1-pD^p)^{n/p + \alpha - 1} x^n = \sum_{k=0}^\infty \binom{\frac{n}{p} + \alpha -1}{k} (-p D^p)^k x^n \nonumber\\
&= \sum_{k=0}^{\floor{n/p}} \binom{\frac{n}{p} + \alpha -1}{k} \frac{n!}{(n-pk)!} (-p)^k x^{n-pk}.
\end{align}

At indices multiples of $p$, we get the nicer formula

\begin{equation}\label{e:lagpn}
\Lag\alpha_{p,pn}(x) = \sum_{k=0}^n \binom{n + \alpha -1}{n-k} \frac{(pn)!}{(pk)!} (-p)^{n-k} x^{pk}.
\end{equation}

It is also possible to find another closed form for $\Lag\alpha_p$.

For $p \in \N$

\[
L_p^{(\alpha)} = e^{-\x D^{p+1} - \alpha pD^p}.
\]

Because $[D^p, \x D^{p+1}] = p (D^p)^2$, by the Berry identity <cit.> and <cit.>,

\[
e^{-\x D^{p+1} - \alpha pD^p} = ((1-p D^p)^{1/p})^{p \alpha} e^{-\x D^{p+1}} = \Lag\alpha_p.
\]

The differential equation characterizing the degenerate associated Laguerre polynomials can be found with <Ref> and the Pincherle derivative

\begin{align*}
(1-pD^p)^\alpha \x &= \x (1-pD^p)^\alpha + ((1-pD^p)^\alpha)' \\
&= \x (1-pD^p)^\alpha - \alpha p^2 D^{p-1} (1-pD^p)^{\alpha-1} \\
&= (\x(1-pD^p) - \alpha p^2 D^{p-1}) (1-pD^p)^{\alpha-1}.
\end{align*}

\begin{align*}
n \Lag\alpha_{p,n}(x)
&= (1-pD^p)^\alpha \x \frac{\Psi_p}{\Psi_p'} L_{p,n}(x) \\
&= (\x(1-pD^p) - \alpha p^2 D^{p-1}) (1-pD^p)^{\alpha-1} D (1 - pD^p) L_{p,n}(x) \\
&= (\x (1-pD^p) - \alpha p^2 D^{p-1}) D \Lag\alpha_{p,n}(x),
\end{align*}

in other words,  $\{\Lag\alpha_{p,n}\}_{n\in\N}$ are solutions to the differential equation

\begin{equation}\label{e:Lpde}
x p f^{(p+1)} + \alpha p^2 f^{(p)} - x f' + n f = 0.
\end{equation}

This generalizes to the associated version what we would have obtained with <ref>.

§.§.§ Laguerre polynomials and Lah numbers

The associated Laguerre polynomials are retrieved through <ref> after setting $p=1$

\begin{equation}
\Lag\alpha_n(x) = \sum_{k=0}^n \binom{n+\alpha-1}{n-k} \frac{n!}{k!} (-1)^{n-k} x^k.
\end{equation}

and with $\alpha = 0$, we obtain the basic Laguerre polynomials

\begin{equation}\label{e:LagLah}
L_n(x) = \sum_{k=0}^n \Lah{n}{k} (-1)^{n-k}x^k,
\end{equation}

where $\Lah{n}{k} \defeq \binom{n - 1}{n-k} \frac{n!}{k!}$ are the unsigned Lah numbers. They are basic for $\Psi \defeq \Psi_1 = D/(1-D)$. We call it the multiderivative as $\Psi = D + D^2 + D^3 + \ldots$ Through $\Delta = \Psi \diamond \nabla$, we obtain

\begin{equation}\label{e:EmmLag}
\varphi = \rho L,
\end{equation}

which is the operational rewording of the following result proved by Ivo Lah when he first introduced Lah numbers [21]

\begin{equation}\label{e:Lahid}
(x)_n = \sum_{k=0}^n  \Lah{n}{k} (-1)^{n-k} \abra{x}_n.
\end{equation}

For an integer $r$, it can be proven by induction that

\begin{equation}
L^r \leadsto \Psi^{[r]} = \frac{D}{1-rD},
\end{equation}

and to compute the associated polynomial sequence of $L^r$, we will use a different technique, i.e., the Niederhausen transform (<Ref>). Because

\[
\Psi^{[-r]} = \frac{D}{1+rD} = D e^{-\log(1+rD)}, \quad \et \quad (-\log(1+rD))^{[-1]} = \frac{-1}{r} \nabla = (-D/r) \diamond \nabla,
\]

we immediately recover from the theorem

\[
\coeff{n}{k}_{L^r} = \binom{n}{k} (-r)^{n-k} \abra{k}_{n-k} = \Lah{n}{k}(-r)^{n-k} = \coeff{n}{k}_L r^{n-k}.
\]

In fact, $r$ can be any number if we consider the fractional exponent of $L$, as discussed at the end of <Ref>

The associated sigma operators of $\Psi$ can be computed with <Ref> (c)

\begin{equation}
\Psi^{-1}_{(a)} = D^{-1}_{(a)} \frac{D}{\Psi} = \I_{(a)} (1-D) = \I_{(a)} + \Ev{a} - 1 .
\end{equation}

Interestingly enough, $(L\sym)^{-1} = \sym^{-1} e^{\x D^2} = e^{-\x D^2} \sym = L \sym$, indicating that $L\sym$ is its own inverse. This is one of the reasons why the literature defines Laguerre polynomials as $L\sym \leadsto - \Psi$. By diverging from this convention, $\Psi$ and $\{L_n\}_{n\in\N}$ remain unitary, and it provides a more natural definition for the degenerate version of Laguerre polynomials.

The Laguerre operator, contrary to the other degenerate Laguerre operators and along $\varphi$ or $\Ca$, is part of the special class of umbral operators, from which we can infer the following.

\[
L \x^n = \sum_{k=0}^n \Lah{n}{k} (-1)^{n-k} \x^k \Lag{k+n}.
\]

We see that $\Psi$ satisfies the fourth form of <Ref>, with $a=1, b=-1, c=-1$, therefore
\[
L\x^n = \sum_{k=0}^n \coeff{n}{k}_L \x^k (1-D)^k (1-D)^n L = \sum_{k=0}^n \Lah{n}{k} (-1)^{n-k} \x^k \Lag{k+n}.
\]

\[
L_n(\lambda x) = \sum_{k=0}^n \Lah{n}{k} \lambda^k (\lambda - 1)^{n-k} L_k(x).
\]

This equality can be proven by two methods, as described in <Ref>. The traditional approach uses the transfer formula [30, 12], but we will illustrate the alternative method.

We compute the coefficient of $X^{(\lambda)} \defeq L^{-1} \str\lambda L$ using <Ref>

\begin{align*}
\coeff{n}{k}_{X^{(\lambda)}}
&= \sum_{j=k}^n \coeff{j}{k}_{L^{-1}} \coeff{n}{j}_{\str\lambda L} = \sum_{j=k}^n \Lah{j}{k} \Lah{n}{j} (-1)^{n-j} \lambda^j \\
&= \frac{n!}{k!} \sum_{j=k}^n \binom{j-1}{k-1} \binom{n-1}{j-1} (-1)^{n-j} \lambda^j = \frac{n!}{k!} \binom{n-1}{k-1} \sum_{j=k}^n  \binom{n-k}{j-k} (-1)^{n-j} \lambda^j \\
&= \Lah{n}{k} \sum_{j=0}^{n-k}  \binom{n-k}{j} (-1)^{n-j-k} \lambda^{j+k} = \Lah{n}{k} \lambda^k (\lambda - 1)^{n-k}.
\end{align*}

We can simply apply $L$ to the formula for $X^{(\lambda)}_n(x)$ to arrive at the desired result.

The classical Laguerre differential equation is obtained in the special case of <ref>

\begin{equation}
x {\Lag\alpha_n}''(x) + (\alpha - x){\Lag\alpha_n} '(x) + n \Lag\alpha_n(x) = 0.
\end{equation}

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