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