# On an Umbral Point of View of the Gaussian and Gaussian-like Functions

^{1}

^{2}

^{*}

*Symmetry*: Applications of Different Mathematical Languages)

## Abstract

**:**

## 1. Introduction

## 2. Gaussian Functions, Lorentzian Functions, and Associated Trigonometric Functions

**Proposition**

**1.**

**Proof.**

**Corollary**

**1.**

**Proposition**

**2.**

**Proof.**

## 3. Higher-Order Gaussian Trigonometric Functions

**Proposition**

**3.**

**Proof.**

**Corollary**

**2.**

**Corollary**

**3.**

**Example**

**1.**

**Remark**

**1.**

**Observation**

**1.**

**Proposition**

**4.**

**Proof.**

**Remark**

**2.**

## 4. New Forms of Gaussian-like Distributions

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

## 5. Applications and Final Comments

**Example**

**7.**

**Example**

**8.**

**Example**

**9.**

**Example**

**10.**

**Example**

**11.**

**Example**

**12.**

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Gaussian trigonometric “circumference”: egg-shaped curve. ${C}_{g}\left(x\right)$ vs. ${S}_{g}\left(x\right)$.

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**MDPI and ACS Style**

Dattoli, G.; Di Palma, E.; Licciardi, S.
On an Umbral Point of View of the Gaussian and Gaussian-like Functions. *Symmetry* **2023**, *15*, 2157.
https://doi.org/10.3390/sym15122157

**AMA Style**

Dattoli G, Di Palma E, Licciardi S.
On an Umbral Point of View of the Gaussian and Gaussian-like Functions. *Symmetry*. 2023; 15(12):2157.
https://doi.org/10.3390/sym15122157

**Chicago/Turabian Style**

Dattoli, Giuseppe, Emanuele Di Palma, and Silvia Licciardi.
2023. "On an Umbral Point of View of the Gaussian and Gaussian-like Functions" *Symmetry* 15, no. 12: 2157.
https://doi.org/10.3390/sym15122157