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arXiv · 2512.04638

On formulas and fractional exponents for umbral operators

Kei Beauduin
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Abstract

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.

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Kei Beauduin. 2026-04-21. On formulas and fractional exponents for umbral operators. https://arxiv.org/abs/2512.04638

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