Hermite polynomial

The $𝑛$ th Hermite polynomial $𝐻 𝑛$ for $𝑛 \in ℕ 0$ is a polynomial of degree $𝑛$ given by the Rodrigues' formula #m/def/fun

𝐻 𝑛 (𝑥) = (- 1) 𝑛 e 𝑥 2 (𝑑 𝑑 𝑥) 𝑛 e - 𝑥 2

in the Physicist's convention and is even or odd depending on the parity of $𝑛$ .

Properties

\int \infty - \infty 𝐻 𝑚 (𝑥) 𝐻 𝑛 (𝑥) e - 𝑥 2 𝑑 𝑥 = \sqrt 𝜋 2 𝑛 𝑛! 𝛿 𝑛 𝑚

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