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On a general identity and a resulting class of umbral operators

Kei Beauduin

TL;DR

The paper extends umbral calculus by proving a universal identity for umbral operators and leveraging it to define and fully characterize a special, shift-invariant subclass. It derives a central identity $φ x^n = \sum_{k=0}^n {n \choose k}_φ x^k B_{n,k}(f^{(1)}(Q), ..., f^{(n-k+1)}(Q)) φ$ and connects it to a functional-equation classification that yields explicit forms for the delta operator $Q = f^{-1}(D)$ and the accompanying operators $U,V$. Through this framework, it demonstrates a range of classical polynomial families (Touchard, Laguerre, Bucchianico) as concrete realizations, providing new operator-based proofs and extended identities (e.g., Spivey-type and Lah-type relations). The results unify operator methods with Bell polynomial structures and illuminate the algebraic structure behind classical umbral sequences.

Abstract

We prove a new universal identity for umbral operators. This motivates the definition of a subclass obeying a simplified identity, which we then fully characterize. The results are illustrated with common examples of the theory of umbral calculus.

On a general identity and a resulting class of umbral operators

TL;DR

The paper extends umbral calculus by proving a universal identity for umbral operators and leveraging it to define and fully characterize a special, shift-invariant subclass. It derives a central identity and connects it to a functional-equation classification that yields explicit forms for the delta operator and the accompanying operators . Through this framework, it demonstrates a range of classical polynomial families (Touchard, Laguerre, Bucchianico) as concrete realizations, providing new operator-based proofs and extended identities (e.g., Spivey-type and Lah-type relations). The results unify operator methods with Bell polynomial structures and illuminate the algebraic structure behind classical umbral sequences.

Abstract

We prove a new universal identity for umbral operators. This motivates the definition of a subclass obeying a simplified identity, which we then fully characterize. The results are illustrated with common examples of the theory of umbral calculus.
Paper Structure (4 sections, 4 theorems, 45 equations)

This paper contains 4 sections, 4 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Demonstration of the identity
  3. A special class of umbral operators
  4. Some examples

Key Result

theorem 1

For all nonnegative integers $n$,

Theorems & Definitions (7)

  • theorem 1
  • lemma 1: beauduin2025
  • proof : of \ref{['t:phixn']}
  • lemma 2
  • proof
  • theorem 2
  • proof