 Hermite polynomial

The physicists' Hermite polynomials areThey are orthogonal on the real line with weight $e^{-x^2}$. The first examples are $1$, $2x$ and $4x^2-2$; they solve the Hermite differential equation $H_n^{\prime \prime }-2x{H}_n^{\prime }+2n{H}_n=0$.

Multiplication by a Gaussian factor converts them into quantum harmonic-oscillator eigenfunctions. The probabilists' normalization instead has weight $e^{-x^2/2}$ and a rescaled variable. Stating the convention prevents factors of two from being hidden in the name. The polynomials are named after Charles Hermite.^[ref]