nLab Hermite polynomial

Contents

Idea

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For $n \in \mathbb{N}$ a natural number, the $n$ th Hermite polynomial is the polynomial in one variable $x$ with coefficients in the real numbers (and as such a naturally regarded as a smooth function $\mathbb{R} \to \mathbb{R}$ on the real line) which is given by the following expression:

(1)

H_n(x) \;\coloneqq\; (-1)^n \exp\left( \tfrac{1}{2} x^2 \right) \frac{d}{d x^n} \exp\left( -\tfrac{1}{2} x^2 \right) \,,

where $\exp(-)$ denotes the exponential function and $d/d x^n$ denotes the $n$ th derivative with respect to $x$ .

Similarly, the Hermite functions are

(2)

\begin{aligned} h_n(x) & \coloneqq\; \frac{ (-1)^n }{ \sqrt{ n! 2^n \sqrt{\pi} } } \exp\left( \tfrac{1}{2} x^2 \right) \frac{d}{d x^n} \exp\left( - x^2 \right) \\ & = \frac{ 1 }{ \sqrt{ n! \sqrt{\pi} } } \, H_n \big( \sqrt{2} x \big) \, \exp \left( -\tfrac{1}{2}x^2 \right) \end{aligned} \,,

Regarded as smooth functions $h_n \colon \mathbb{R} \to \mathbb{R}$ on the real line, the Hermite functions (2) form an orthonormal basis of square-integrable functions with respect to the 2-norm, in that the integration over their product satisfies

(3)

\int_{\mathbb{R}^1} h_{n}(x) h_{m}(x) \, d x \;=\; \delta_{n,m}

(where on the right we have the Kronecker delta).

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(4)

\frac{d}{d x} h_n(x) \;=\; \sqrt{ \tfrac {n} {2} } h_{n-1}(x) - \sqrt{ \tfrac {n-1} {2} } h_{n+1}(x)

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