6 papers
The indicator diagram theory for maximal type entire functions and the characterization of analytic continuability
Kei Beauduin
We extend Pólya's indicator diagram theory to encompass entire functions of order at most 1, allowing functions of maximal type. To do so, we introduce an extension of the complex…
A general identity for umbral operators and a special subclass
Kei Beauduin
We prove a new universal identity for umbral operators. This motivates the definition of a subclass satisfying a simplified identity, which we fully characterize. The results are i…
On formulas and fractional exponents for umbral operators
Kei Beauduin
We present a new formula for umbral operators that yields three main insights. First, it makes explicit a connection between umbral calculus and iteration theory. Second, it leads…
Explicit Expressions for Iterates of Power Series
Kei Beauduin
In this paper, we present several formulas for both the discrete and fractional iterates of an invertible power series , using a new unifying approach based on umbral calculus.…
Operational Umbral Calculus
Kei Beauduin
In this paper, we investigate the power of nearly purely operational techniques in the study of umbral calculus. We present a concise reconstruction of the theory based on a system…
Old and new powerful tools for the normal ordering problem and noncommutative binomials
Kei Beauduin
In this paper, we derive formal general formulas for noncommutative exponentiation and the exponential function, while also revisiting an unrecognized, and yet powerful theorem. Th…