# Voigt Transform and Umbral Image

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^{2}

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## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

**Proposition**

**1.**

**Proof.**

**Corollary**

**1.**

**Proposition**

**2**

**.**The relevant NOH-function integral representation can be recast as:

**Proof.**

**Example**

**1.**

**Example**

**2.**

**Proposition**

**3**

**.**$\forall x,y\in \mathbb{R},\forall n\in \mathbb{N},$ the Laguerre polynomials can be represented by an umbral image using the Newton binomial:

**Corollary**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

## 2. Voigt Functions, Hermite Functions, and Generalized Forms

**Definition**

**2.**

**Observation**

**1.**

**Proposition**

**4.**

**Proof.**

**Corollary**

**3.**

**Definition**

**3**

**.**$\forall x,z\in \mathbb{R}$, $\forall y\in {\mathbb{R}}^{+}$, we introduce the Voigt (V-)transform of a function $f\left(z\right)$:

**Proposition**

**5.**

**Proof.**

**Corollary**

**4.**

**Example**

**6.**

**Remark**

**1.**

**Example**

**7.**

**Example**

**8**

**.**In this type of device, the growth of the electromagnetic field is ruled by an integro-differential equation, which can be cast in the form [10]:

## 3. Final Comments and Applications

**Example**

**9.**

**Example**

**10.**

**Example**

**11**

**.**To study the $GFB$, we introduce the higher order Hermite polynomials, also called lacunary Hermite polynomials (or Kampé de Fériet or Gould Hopper polynomials), defined through the operational identity (see Equation (4.8.4) and the following in [5]):

**Example**

**12**

**.**Within the present context, the Gauss–Weierstrass transform can be viewed as being generated from the operational identity:

**Example**

**13**

**.**The same point of view can be followed to introduce the Laguerre transform, which can be derived from the identity:

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Licciardi, S.; Pidatella, R.M.; Artioli, M.; Dattoli, G.
Voigt Transform and Umbral Image. *Math. Comput. Appl.* **2020**, *25*, 49.
https://doi.org/10.3390/mca25030049

**AMA Style**

Licciardi S, Pidatella RM, Artioli M, Dattoli G.
Voigt Transform and Umbral Image. *Mathematical and Computational Applications*. 2020; 25(3):49.
https://doi.org/10.3390/mca25030049

**Chicago/Turabian Style**

Licciardi, Silvia, Rosa Maria Pidatella, Marcello Artioli, and Giuseppe Dattoli.
2020. "Voigt Transform and Umbral Image" *Mathematical and Computational Applications* 25, no. 3: 49.
https://doi.org/10.3390/mca25030049