For questions related to Hermite polynomials. Hermite polynomials are polynomials of the form $\operatorname{H}{n}\left( x \right) = \left( -1 \right)^{n} \cdot e^{x^{2}} \cdot \frac{\operatorname{d}^{n}}{\operatorname{d}x^{n}} e^{-x^{2}}$ or $\operatorname{He}{n}\left( x \right) = 2^{\frac{n}{2}} \cdot \operatorname{H}{n}\left( \sqrt{2} \cdot x \right)$ for $n \in \mathbb{N}{0}$.
Hermite polynomials are a class of orthogonal polynomials that are defined as follows: \begin{align*} \text{Probabilist's Hermite polynomials}&\text{: } \operatorname{He}_{n}\left( x \right) = \left( -1 \right)^{n} \cdot e^{\frac{x^{2}}{2}} \cdot \frac{\operatorname{d}^{n}}{\operatorname{d}x^{n}} e^{-\frac{x^{2}}{2}}\\ \text{Physicist's Hermite polynomials}&\text{: } \operatorname{H}_{n}\left( x \right) = \left( -1 \right)^{n} \cdot e^{x^{2}} \cdot \frac{\operatorname{d}^{n}}{\operatorname{d}x^{n}} e^{-x^{2}}\\ \end{align*}
These polynomials are orthogonal with respect to the weight function: \begin{align*} \int\limits_{-\infty}^{\infty} \operatorname{He}_{m}\left( x \right) \cdot \operatorname{He}_{n}\left( x \right) \cdot e^{-\frac{x^{2}}{2}}\, \operatorname{d}x &= \sqrt{2 \cdot \pi} \cdot n! \cdot \delta_{nm}\\ \int\limits_{-\infty}^{\infty} \operatorname{H}_{m}\left( x \right) \cdot \operatorname{H}_{n}\left( x \right) \cdot e^{-x^{2}}\, \operatorname{d}x &= \sqrt{\pi} \cdot 2^{n} \cdot n! \cdot \delta_{nm}\\ \end{align*}
Read more about hermite polynomials and their properties here.
