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Hermite Polynomials 📂Functions

Hermite Polynomials

Description

Hermite Polynomials are defined in several ways as follows.

As solutions to a differential equation

Hermite polynomials are defined as the solutions to the following Hermite Differential Equation.

$$ y^{\prime \prime} -2xy^{\prime} +2ny=0,\quad n=0,1,2,\cdots $$

Rodrigues’ formula

The following function $H_{n}$ is called the Hermite polynomial.

$$ H_{n}(x)=(-1)^{n}e^{x^{2}}\frac{ d ^{n}}{ dx^{n} }e^{-x^{2}} $$

This is known as the Rodrigues’ formula. Meanwhile, the above function is referred to as the physicist’s Hermite polynomials, and the form below is called the probabilist’s Hermite polynomials.

$$ H_{e_{n}} := (-1)^{n} e^{{x^2} \over {2}} {{d^{n}} \over {dx^{n}}} e^{- {{x^2} \over {2}}} $$

Explanation

By definition, $H_{n}$ is indeed a polynomial ‘function’, but it is conventionally called Hermite ‘polynomial’. This is not just in Korean, as the English expression is also Hermite polynomial, not polynomial function.

The first few Hermite polynomials are as follows:

$$ \begin{align*} H^{0}(x) &= 1 \\ H^{1}(x) &= 2x \\ H^{2}(x) &= 4x^{2}-2 \\ H^{3}(x) &= 8x^{3}-12x \\ H^{4}(x) &= 16x^{4}-48x^{2}+12 \\ H^{5}(x) &= 32x^{5}-160x^{3}+120x \\ &\vdots \end{align*} $$

Properties

Orthogonality

Hermite polynomials are orthogonal with respect to the weight function $w(x)=e^{-x^{2}}$ over the interval $(-\infty, \infty)$. (Link)

$$ \braket{ H_{n} | H_{m} }_{e^{-x^{2}}} =\int_{-\infty}^{\infty}e^{-x^{2}}H_{n}(x)H_{m}(x)dx=\sqrt{\pi}2^{n}n!\delta_{nm} $$

Recurrence relation

Hermite polynomials satisfy the following recurrence relation. (Link)

$$ \begin{align*} H_{n}^{\prime}(x) &= 2nH_{n-1}(x) \\ H_{n+1}(x) &= 2xH_{n}(x)-2nH_{n-1}(x) \\ &= 2xH_{n}(x)-H_{n}^{\prime}(x) \nonumber \end{align*} $$

Generating function

The generating function of Hermite polynomials is as follows. (Link)

$$ \Phi (x,t)=\sum \limits _{n=0}^{\infty} \frac{H_{n}(x)}{n!}t^{n}= e^{2xt-t^{2}} $$