# Dual Numbers and Operational Umbral Methods

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## Abstract

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## 1. Introduction

## 2. Higher-Order Dual Numbers

## 3. Umbral-Type Methods and Dual Numbers

#### 3.1. Dual Shifted Gaussians

#### 3.2. Another Form of Dual Gaussian

#### 3.3. Examples From Symbolic Calculus

#### 3.4. Umbral Image Type Techniques

## 4. Dual Numbers and Solution of Heat- and Schrödinger-Type Equations

## 5. Weyl Formula and Modified Hermite Polynomials

## 6. Final Comments

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Behr, N.; Dattoli, G.; Lattanzi, A.; Licciardi, S. Dual Numbers and Operational Umbral Methods. *Axioms* **2019**, *8*, 77.
https://doi.org/10.3390/axioms8030077

**AMA Style**

Behr N, Dattoli G, Lattanzi A, Licciardi S. Dual Numbers and Operational Umbral Methods. *Axioms*. 2019; 8(3):77.
https://doi.org/10.3390/axioms8030077

**Chicago/Turabian Style**

Behr, Nicolas, Giuseppe Dattoli, Ambra Lattanzi, and Silvia Licciardi. 2019. "Dual Numbers and Operational Umbral Methods" *Axioms* 8, no. 3: 77.
https://doi.org/10.3390/axioms8030077