# Monomiality and a New Family of Hermite Polynomials

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## Abstract

**:**

## 1. Introduction

**Properties**

**1.**

- (a)
- (b)
- (c)
- It is possible to univocally define a polynomial set such that:$$\left(i\right)\phantom{\rule{0.277778em}{0ex}}{p}_{0}\left(x\right)=1,\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\left(ii\right)\phantom{\rule{0.277778em}{0ex}}\widehat{P}{p}_{0}\left(x\right)=0,\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\left(iii\right)\phantom{\rule{0.277778em}{0ex}}{p}_{n}\left(x\right)={\widehat{M}}^{n}1,$$

- (d)
- $$\widehat{M}{p}_{n}\left(x\right)={\widehat{M}}^{n+1}1={p}_{n+1}\left(x\right),$$
- (e)
- $$\widehat{P}{p}_{n}\left(x\right)=\widehat{P}{\widehat{M}}^{n}1=n{p}_{n-1}\left(x\right)$$

**Proof.**

**Remark**

**1.**

**Corollary**

**1.**

**Corollary**

**2.**

## 2. Quasi-Hermite and Appéll Sequences

**Definition**

**1.**

**Properties**

**2.**

**Proof.**

- (i)
- From property $\left(2\right)$, we write$$\left(1+{\displaystyle \frac{y}{N}}{\partial}_{x}^{2}\right){H}_{n+1}=\left(1+{\displaystyle \frac{y}{N}}{\partial}_{x}^{2}\right)x{H}_{n}+2y{\partial}_{x}{H}_{n}$$
- (ii)
- $${H}_{n+1}+{\displaystyle \frac{y}{N}}n(n+1){H}_{n-1}=x{H}_{n}+{\displaystyle \frac{y}{N}}\left(2n{H}_{n-1}+n(n-1)x{H}_{n-2}\right)+2ny{H}_{n-1}$$
- (iii)
- $$\begin{array}{cc}\hfill {H}_{n+1}-x{H}_{n}-2ny{H}_{n-1}& ={\displaystyle \frac{y}{N}}n\left(\left(2-(n+1)\right){H}_{n-1}+(n-1)x{H}_{n-2}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={\displaystyle \frac{y}{N}}n(n-1)\left(x{H}_{n-2}-{H}_{n-1}\right).\hfill \end{array}$$

**Proposition**

**1.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Corollary**

**4.**

**Example**

**1.**

**Observation**

**1.**

**Observation**

**2.**

## 3. Multivariable QHP

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

## 4. Final Comments

**Proposition**

**2.**

**Corollary**

**5.**

- 1.
- We apply the Gauss–Weierstrass transform [22] to write$$\begin{array}{cc}\hfill {e}^{s\frac{\mid y\mid}{N}{\partial}_{x}^{2}}f\left(x\right)& ={\displaystyle \frac{1}{2\sqrt{\pi \phantom{\rule{0.277778em}{0ex}}s\frac{\mid y\mid}{N}}}}{\int}_{-\infty}^{\infty}exp\left\{-{\displaystyle \frac{{(x-\xi )}^{2}}{4\phantom{\rule{0.277778em}{0ex}}s\frac{\mid y\mid}{N}}}\right\}f\left(\xi \right)d\xi \hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{0.277778em}{0ex}}={\displaystyle \frac{1}{2\sqrt{\pi \phantom{\rule{0.277778em}{0ex}}s\frac{\mid y\mid}{N}}}}{\int}_{-\infty}^{\infty}{e}^{-\frac{1}{4\phantom{\rule{0.277778em}{0ex}}s\frac{\mid y\mid}{N}}{\xi}^{2}}{e}^{\frac{x}{2s\frac{\mid y\mid}{N}}\xi}{e}^{-\frac{{x}^{2}}{4s\frac{\mid y\mid}{N}}}d\xi .\hfill \end{array}$$
- 2.
- We use the two variable Hermite-generating functions (we have ${\sum}_{n=0}^{\infty}\frac{{t}^{n}}{n!}{H}_{n}(x,y)={e}^{xt+y{t}^{2}}$) [38] to write$${e}^{s\frac{\mid y\mid}{N}{\partial}_{x}^{2}}f\left(x\right)={\displaystyle \frac{1}{2\sqrt{\pi \phantom{\rule{0.277778em}{0ex}}s\frac{\mid y\mid}{N}}}}\sum _{n=0}^{\infty}{\displaystyle \frac{{x}^{n}}{n!}}{\int}_{-\infty}^{\infty}{H}_{n}\left({\displaystyle \frac{\xi}{2s\frac{\mid y\mid}{N}}},-{\displaystyle \frac{1}{4s\frac{\mid y\mid}{N}}}\right){e}^{-\frac{{\xi}^{2}}{4\phantom{\rule{0.277778em}{0ex}}s\frac{\mid y\mid}{N}}}f\left(\xi \right)d\xi .$$
- 3.
- We insert the result of Equation (51) into Equation (49) and compare the similar x powers, thus eventually finding$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {a}_{n}={\displaystyle \frac{1}{\Gamma \left(N\right)n!\phantom{\rule{0.277778em}{0ex}}2\sqrt{\pi {\displaystyle \frac{\mid y\mid}{N}}}}}{\int}_{0}^{\infty}{s}^{N-\frac{3}{2}}{e}^{-s}\phantom{\rule{0.277778em}{0ex}}{}_{n}{G}_{y,N}\left(s\right)ds,\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {}_{n}{G}_{y,N}\left(s\right)={\int}_{-\infty}^{\infty}{H}_{n}\left({\displaystyle \frac{\xi}{2s\frac{\mid y\mid}{N}}},-{\displaystyle \frac{1}{4s\frac{\mid y\mid}{N}}}\right){e}^{-\frac{{\xi}^{2}}{4\phantom{\rule{0.277778em}{0ex}}s\frac{\mid y\mid}{N}}}f\left(\xi \right)d\xi .\hfill \end{array}$$

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Dattoli, G.; Licciardi, S.
Monomiality and a New Family of Hermite Polynomials. *Symmetry* **2023**, *15*, 1254.
https://doi.org/10.3390/sym15061254

**AMA Style**

Dattoli G, Licciardi S.
Monomiality and a New Family of Hermite Polynomials. *Symmetry*. 2023; 15(6):1254.
https://doi.org/10.3390/sym15061254

**Chicago/Turabian Style**

Dattoli, Giuseppe, and Silvia Licciardi.
2023. "Monomiality and a New Family of Hermite Polynomials" *Symmetry* 15, no. 6: 1254.
https://doi.org/10.3390/sym15061254