On formulas and fractional exponents for umbral operators

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.