On formulas and fractional exponents for umbral operators

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 naturally to a definition of fractional exponents of umbral operators. Third, its proof synthesizes a broad range of existing results in operational calculus and highlights their combined effectiveness. As an illustration, we obtain a new and natural extension of the Laguerre polynomials.