On formulas and fractional exponents for umbral operators
Kei Beauduin
TL;DR
The paper develops a unified, operator-centric framework for umbral calculus by representing general umbral operators as $\phi = e^{\text{\ttfamily x} V(D)}$ and establishing an explicit form for $V$ via iteration theory. It integrates classical results (Pincherle, Garsia–Joni, Steffensen, Kurbanov–Maksimov) with modern tools such as the L-transform and fractional iteration to derive a concise expression $\phi = e^{\text{\ttfamily x} \mathrm{itlog}(f)(D)}$, enabling fractional powers $\phi^s$ and a converse construction from $V$ to a generating function $f$. The framework yields a rigorous path to fractional exponents of umbral operators and demonstrates new Laguerre-type polynomial families (degenerate Laguerre) as natural applications. Overall, the work deepens the conceptual bridge between umbral calculus, operator theory, and iteration theory, expanding the operational calculus toolkit for special polynomials and their generalizations.
Abstract
This study presents a new formula for umbral operators which provides three key insights. First, it clarifies a connection between umbral calculus and iteration theory. Second, it paves the way for a definition of fractional exponents of umbral operators. And lastly, its proof synthesizes a multitude of existing operational calculus results that demonstrates a new level of effectiveness in the field. We demonstrate its application through a new and natural extension of the Laguerre polynomials.