Hermite Polynomials
Definition
Probabilist’s Hermite Polynomial
$$ H_{e_{n}} := (-1)^{n} e^{{x^2} \over {2}} {{d^{n}} \over {dx^{n}}} e^{- {{x^2} \over {2}}} $$
Physicist’s Hermite Polynomial
$$ H_{n} := (-1)^{n} e^{x^2} {{d^{n}} \over {dx^{n}}} e^{-x^2} $$
Basic Properties
Hermite polynomials are used in two forms, having a relationship as shown in $H_{n} (x) = 2^{{n} \over {2}} H_{e_{n}} \left( \sqrt{2} x \right)$.
Recurrence Relation
- [0]: $$H_{n+1} (x) = 2x H_{n} (x) - H_{n} ' (X)$$
Orthogonal Set
- [1] Inner product of functions: Given the weight for $\displaystyle \left<f, g\right>:=\int_a^b f(x) g(x) w(x) dx$ as $w$, then $\displaystyle w(x) := e^{-x^2}$ forms an orthogonal set.
Explanation
The physicist’s Hermite polynomials for $n = 0, \cdots , 3$ are expressed as follows:
$$ \begin{align*} H_{0} (x) =& 1 \\ H_{1} (x) =& 2x \\ H_{2} (x) =& 4 x^2 - 2 \\ H_{3} (x) =& 8 x^3 - 12x \end{align*} $$
The probabilist’s Hermite polynomials are also defined as solutions to the Hermite differential equation $y’’ - x y ' + 2 n y = 0$.
The closed form for the Hermite nodes $x_{k}$ that satisfy $H_{n} ( x_{k} ) = 0$ is unfortunately unknown, and is still being computed numerically.