Fractional Exponents of Umbral Operators

• Fractional exponents of umbral operators are defined via iterative logarithm and binomial series, generalizing integer power concepts to non-integer settings.
• The framework yields a one-parameter semigroup with entire analytic dependence, enabling spectral decompositions and explicit combinatorial coefficient evaluations.
• Applications span fractional differential equations, analytic continuation of special functions, and combinatorial identities, bridging operator theory with advanced calculus.

A fractional exponent of an umbral operator generalizes the classical notion of operator powers from integer to complex or real exponents, allowing for a functional calculus on umbral objects central to combinatorics, operational calculus, and the spectral theory of linear operators. This framework systematically extends the action, properties, and applications of umbral operators—commonly appearing in the theory of polynomial sequences of binomial type, difference and shift operators, and analytic functionals—beyond the integral context, revealing deep connections to iteration theory, analytic continuation, fractional summation, and numerous special functions.

1. Formal Construction and Basic Properties

Let $\phi$ be an invertible umbral operator acting on a polynomial ring $\mathbb{C}[x]$, typically induced by an invertible power series $f(t)$ with $a_1 \neq 0$. According to operational umbral calculus (Beauduin, 23 Jul 2024, Beauduin, 4 Dec 2025), the fractional exponent $\phi^s$ for $s \in \mathbb{C}$ is defined via either of the following equivalent frameworks:

• Exponential of the Iterative Logarithm:

$$\phi^s = \exp\left(s\,x\;\mathrm{itlog}\,f(D)\right)$$

where

$\mathrm{itlog}\,f(t)$

is Écalle's iterative logarithm of $f$, satisfying $$\mathrm{itlog}\,f\circ f = f'(0) \cdot \mathrm{itlog}\,f$$. The operator $\phi^s$ then acts as the unique umbral operator corresponding to the compositional $s$-th iterate of $f$.

• Binomial Series Expansion:

$$\phi^s = \sum_{k=0}^\infty \binom{s}{k} (\phi - I)^k$$

interpreted formally or in operator topology. This expansion directly generalizes the classical expression $(1+A)^s$ for a (possibly non-commuting) additive operator $A$.

• Indicator Function Calculus:

If $\tilde\phi(t)$ is the operator's indicator, then

$\phi^s = \tilde\phi(t)^s \Big|_{t \mapsto D}$

which defines fractional powers for all shift-invariant operators $\phi$.

This construction yields a one-parameter group $\{\phi^s\}_{s\in\mathbb{C}}$, obeying $\phi^{s}\phi^{r} = \phi^{s+r}$, with analytic dependence on $s$ and a natural infinitesimal generator $\log \phi = x\,\mathrm{itlog}\,f(D)$ (Beauduin, 4 Dec 2025, Beauduin, 23 Jul 2024).

2. Analytic and Algebraic Structure

The group $\{\phi^s: s \in \mathbb{C}\}$ possesses a rich algebraic and analytic structure:

• Semigroup Law and Entire Dependence: $\phi^s \phi^t = \phi^{s+t}$, and all map coefficients of $\phi^s x^n$ are entire in $s$ (Beauduin, 23 Jul 2024, Beauduin, 4 Dec 2025).
• Spectral-Type Integral Formulae: For suitable $\phi$, classical operator theory supplies

$$\phi^s = \frac{1}{2\pi i} \oint_\Gamma z^s (zI - \phi)^{-1} dz$$

and

$$\phi^{-s} = \frac{1}{\Gamma(s)} \int_0^\infty t^{s-1} e^{-t\phi}\,dt$$

reminiscent of the spectral definition of fractional powers for self-adjoint operators (Beauduin, 4 Dec 2025, Thangavelu, 2018).

• Commutation with Associated Delta Operators: The delta operator $Q^{[s]} = f^{-s}(D)$ satisfies the intertwining relation

$Q^{[t]} \phi^s = \phi^s Q^{[t-s]}$

essential for translation-invariant frameworks and polynomial sequence enumeration (Beauduin, 4 Dec 2025).

• Explicit Matrix and Coefficient Expansions: The coefficients $c_{n,k}(s)$ in $\phi^s x^n = \sum_{k=0}^n c_{n,k}(s)x^k$ are given by explicit combinatorial sums involving binomial transforms and indicator powers, which generalize the Jabotinsky matrix for all $s \in \mathbb{C}$ (Beauduin, 23 Jul 2024, Beauduin, 4 Dec 2025).

3. Analytic Evaluation and Functional Calculus

The analytic functional calculus for umbral operators is based on a generalized evaluation map:

• For an umbra $A$ (a generating pair $(\mathcal{A},\Omega_{a,b})$), the "fractional power" $A^z$ is defined by analytic continuation:

$$A^z := \mathrm{eval}_A\left(w \mapsto e^{z \ln w}\right)$$

implemented as a contour integral using the generalized Fourier transform $\widehat{\mathcal{A}}(\xi)$:

$$A^z = \int_{-\infty - i t}^{+\infty - i t} \widehat{\mathcal{A}}(\xi) (i\,\xi)^z d\xi, \quad t \in (\alpha,\beta)$$

This definition ensures $A^n$ coincides with the $n$th moment (Taylor coefficient) of $\mathcal{A}$ if $n \in \mathbb{N}$, and $z \mapsto A^z$ is entire under natural analyticity conditions (Qian, 2021).

• For polynomial operators with explicit spectral decompositions, as in the Hermite operator $H = -\Delta + |x|^2$, the fractional power $H^s$ is given by

$H^s = \sum_{k=0}^\infty (2k+n)^s P_k$

where $P_k$ are spectral projections, and inverse powers admit integral representations using the Gamma function:

$$H^{-s} = \frac{1}{\Gamma(s)} \int_0^\infty t^{s-1} e^{-t H} dt$$

(Thangavelu, 2018).

4. Illustrative Special Cases and Applications

Numerous classical operators and special functions are encompassed within this framework:

• Shift and Difference Operators: For the forward shift $E^a$, the fractional power is $e^{s a D}$, corresponding to the $s$-fold shift $f(x) \mapsto f(x + s a)$. For the forward difference $\Delta = E - 1$, $\Delta^s$ implements a fractional difference, with the corresponding binomial expansion on monomials:

$$\Delta^\alpha x^m = \sum_{k=0}^m \binom{m}{k} \binom{\alpha}{k} (-1)^k x^{m-k}$$

(Qian, 2021, Beauduin, 23 Jul 2024).

• Bernoulli Umbra: For the Bernoulli umbra $B$ with $\mathcal{A}_B(z) = z e^z / (e^z - 1)$, the fractional moment is given by the contour integral and results in the explicit formula $B^z = -z \zeta(1-z)$, linking the umbral calculus to fractional zeta and polylogarithm structures (Qian, 2021).
• Mittag-Leffler Functions and Operator Exponentials: The Mittag-Leffler function $E_{\alpha,1}(t\hat{O})$ arises as the fractional exponential of an umbral operator via the expansion

$$E_{\alpha,1}(t\hat{O}) = \sum_{n=0}^{\infty} \frac{(t\hat{O})^n}{\Gamma(\alpha n + 1)}$$

This underpins solutions of fractional partial differential equations, such as

$F(x,t) = E_{\alpha,1}(t^\alpha \partial_x^2)f(x)$

for fractional diffusion and quantum evolution (Dattoli et al., 2017).

• Laguerre and Abel Operators: Explicit fractional exponents of generalized Laguerre and Abel umbral operators provide fractional analogues of these classical polynomial systems; for instance:

$$L^s(x^n) = \sum_{k=0}^n \binom{s}{n-k} \binom{n}{k} (-1)^{n-k} x^k$$

(Beauduin, 23 Jul 2024, Beauduin, 4 Dec 2025).

5. Structural and Theoretical Implications

Fractional exponents of umbral operators unify several analytic and combinatorial phenomena:

• Iteration Theory: The use of Écalle's iterative logarithm and exponential directly ties the operator calculus to compositional iteration, extending operator monoids to actions parametrized by $\mathbb{C}$ and not just $\mathbb{N}$ (Beauduin, 4 Dec 2025).
• Spectral and Symbolic Calculi: For self-adjoint and umbrally-diagonalizable operators, the fractional calculus passes through well-defined spectral decompositions and Weyl symbol expansions, enabling explicit estimates and integral representations critical for fractional PDEs and harmonic analysis (Thangavelu, 2018).
• Fractional Summation and Difference Theory: The framework connects to axiomatized fractional summation (Müller–Schleicher), with umbral integrals reproducing known analytic continuation results for fractional sums and differences, including precise regularity and commutativity conditions for differentiation and summation (Qian, 2021).
• Operator Algebra and Jabotinsky Matrices: Fractional powers induce binomial-type transforms on coefficient matrices representing operators, facilitating combinatorial and analytic manipulations across the entire semigroup $\{\phi^s\}$ (Beauduin, 23 Jul 2024).

6. Limitations and Generalizations

Analyticity—in particular continuation into vertical strips for generating functions and the test functions—is a key technical limitation. Operators whose generating functions lack the appropriate decay or regularity are excluded from this framework (Qian, 2021). Extension to higher umbral products, operator pseudoinverses, and the incorporation of functional measure moments (e.g., Dirichlet series umbrae) are possible through the same analytic machinery but demand additional technical care (Beauduin, 4 Dec 2025, Beauduin, 23 Jul 2024).

Further, in the context of noncommuting operators, the fractional-exponential umbral calculus admits parallel analogues of Baker–Campbell–Hausdorff expansions and operator ordering rules, consistently generalizing standard exponential semigroup structures (Dattoli et al., 2017).

7. Notable Applications

Fractional umbral calculus underlies:

• Analytic continuation of special function and zeta values (e.g., Bernoulli umbra and values of the Riemann zeta function at fractional arguments) (Qian, 2021).
• Explicit representation for solutions to fractional heat and Schrödinger equations via operator-valued Mittag-Leffler functions, yielding fractional Poisson statistics and anomalous diffusion (Dattoli et al., 2017).
• Generalization of combinatorial and operational identities within polynomial sequence and Riordan array theory, with direct implications for summation to non-integral bounds and general identities involving binomial-type polynomial systems (Beauduin, 4 Dec 2025, Beauduin, 23 Jul 2024).
• Harmonically, closed-form symbol expansions for fractional powers of classical operators using Laguerre and Hermite generating functions (Thangavelu, 2018).

The modern operational perspective positions fractional exponents of umbral operators as a robust, unifying analytic-combinatorial instrument, extending both classical operator theory and the combinatorics of polynomial sequences into the continuous and complex domains.