Special functions MOC

Hermite polynomial

The 𝑛th Hermite polynomial 𝐻𝑛 for 𝑛 ∈ℕ0 is a polynomial of degree 𝑛 given by the Rodrigues’ formula fun

𝐻𝑛(𝑥)=(−1)𝑛e𝑥2(𝑑𝑑𝑥)𝑛e−𝑥2

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

Properties

1. The Hermite polynomials satisfy the orthonormality condition

∫∞−∞𝐻𝑚(𝑥)𝐻𝑛(𝑥)e−𝑥2𝑑𝑥=√𝜋2𝑛𝑛!𝛿𝑛𝑚

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