Hermite (probabilist)

The Hermite polynomials $\{H_n\}_{n=0}^{\infty}$ are a sequence of orthogonal polynomials that satisfy the differential equation $$\left\{ \begin{array}{ll}

\end{array} \right.$$

Properties

Theorem: The Hermite polynomials $H_n$ satisfy the Rodrigues' formula $$H_n(t) = (-1)^ne^{x^2}\dfrac{d^n}{dx^n}e^{-x^2}.$$

Proof: █

Theorem: (Generating function) The Hermite polynomials obey $$e^{2tx-t^2} = \displaystyle\sum_{k=0}^{\infty} \dfrac{H_k(x)t^n}{n!}.$$

Proof: █

Theorem: (Orthogonality) The Hermite polynomials obey $$\displaystyle\int_{-\infty}^{\infty} e^{-x^2}H_n(x)H_m(x)dx=\left\{ \begin{array}{ll} 0 &; m \neq n \\ 2^nn!\sqrt{\pi} &; m=n \end{array} \right..$$

Proof: █

Theorem: $H_n(x)$ is an even function for even $n$ and an odd function for odd $n$.

Proof: █