Editing Hermite polynomials (section)

=== Other properties ===
The addition theorem, or the summation theorem, states that<ref name=":1">{{Cite web |title=DLMF: §18.18 Sums ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials |url=https://dlmf.nist.gov/18.18 |access-date=2025-03-18 |website=dlmf.nist.gov}}</ref><ref>{{Cite book |last1=Gradshteĭn |first1=I. S. |title=Table of integrals, series, and products |last2=Zwillinger |first2=Daniel |date=2015 |publisher=Elsevier, Academic Press is an imprint of Elsevier |isbn=978-0-12-384933-5 |edition=8 |location=Amsterdam ; Boston}}</ref>{{Pg|location=8.958}}<math display="block">\frac{\left(\sum_{k=1}^r a_k^2\right)^{\frac{n}{2}}}{n!} H_n\left(\frac{\sum_{k=1}^r a_k x_k}{\sqrt{\sum_{k=1}^r a_k^2}}\right)=\sum_{m_1+m_2+\ldots+m_r=n, m_i \geq 0} \prod_{k=1}^r\left\{\frac{a_k^{m_k}}{m_{k}!} H_{m_k}\left(x_k\right)\right\}  </math>for any nonzero vector <math>a_{1:r}</math>.

The multiplication theorem states that<ref name=":1" /><math display="block">H_{n}\left(\lambda x\right)=\lambda^{n}\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{{\left(-n\right)_{2\ell}}}{\ell!}(1-\lambda^{-2})^{\ell}H_{n-2\ell}\left(x\right)</math>for any nonzero <math>\lambda</math>.

Feldheim formula<ref name=":2">Feldheim, Ervin. "Développements en série de polynômes d’Hermite et de Laguerrea l’aide des transformations de Gauss et de Hankel." Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 435 (1940). Part [https://dwc.knaw.nl/DL/publications/PU00017406.pdf I], [https://dwc.knaw.nl/DL/publications/PU00017407.pdf II], [https://dwc.knaw.nl/DL/publications/PU00017420.pdf III]</ref>{{Pg|location=Eq 46}}<math display="block">\begin{aligned}
\frac{1}{\sqrt{a \pi}} & \int_{-\infty}^{+\infty} e^{-\frac{x^2}{a}} H_m\left(\frac{x+y}{\lambda}\right) H_n\left(\frac{x+z}{\mu}\right) d x \\
& = \left(1-\frac{a}{\lambda^2}\right)^{\frac{m}{2}}\left(1-\frac{a}{\mu^2}\right)^{\frac{n}{2}} 
\sum_{r=0}^{\min (m, n)}  r!\binom{m}{r}\binom{n}{r} \left(\frac{2 a}{\sqrt{\left(\lambda^2-a\right)\left(\mu^2-a\right)}}\right)^r H_{m-r}\left(\frac{y}{\sqrt{\lambda^2-a}}\right) H_{n-r}\left(\frac{z}{\sqrt{\mu^2-a}}\right)
\end{aligned}</math>where <math>a \in \mathbb C</math> has a positive real part. As a special case,<ref name=":2" />{{Pg|location=Eq 52}}<math display="block">\frac{1}{\sqrt{\pi}} \int_{-\infty}^{+\infty} e^{-t^2} H_m(t \sin \theta+v \cos \theta) H_n(t \cos \theta-v \sin \theta) d t =(-1)^n \cos ^m \theta \sin ^n \theta H_{m+n}(v)</math>