Hermite polynomial

In mathematics and physics, Hermite polynomials form a well-known class of orthogonal polynomials. In quantum mechanics they appear as eigenfunctions of the harmonic oscillator and in numerical analysis they play a role in Gauss-Hermite quadrature. The functions are named after the French mathematician Charles Hermite (1822–1901).

• See Addendum for a table of Hermite polynomials through n = 12.

Orthonormality

The Hermite polynomials H_n(x) are orthogonal in the sense of the following inner product:

$\left(H_{n'},H_{n}\right)\equiv \int _{-\infty }^{\infty }H_{n'}(x)H_{n}(x)\;e^{-x^{2}}\,\mathrm {d} x=\delta _{n'n}\,h_{n}.$

That is, the polynomials are defined on the full real axis and have weight w(x) = exp(−x²). Their orthogonality is expressed by the appearance of the Kronecker delta δ_n'n. The normalization constant is given by

$N_{n}\equiv {\sqrt {\frac {1}{h_{n}}}}=\left({\frac {1}{\pi }}\right)^{1/4}\,{\frac {1}{\sqrt {2^{n}\,n!}}}.$

Normalization is to unity

$N_{n}^{2}\;\int _{-\infty }^{\infty }H_{n}(x)H_{n}(x)\;e^{-x^{2}}\,\mathrm {d} x=1.$

The polynomials N_nH_n(x) are orthonormal, which means that they are orthogonal and normalized to unity.

Explicit expression

$H_{n}(x)=n!\,\sum _{m=0}^{\lfloor N/2\rfloor }\;(-1)^{m}{\frac {1}{m!(n-2m)!}}\,(2x)^{n-2m}$

here $\scriptstyle \lfloor N/2\rfloor =N/2$ if N even and $\scriptstyle \lfloor N/2\rfloor =(N-1)/2$ if N odd.

Recursion relation

Orthogonal polynomials can be constructed recursively by means of a Gram-Schmidt orthogonalization pocedure. This procedure yields the following relation

$H_{n+1}(x)=2xH_{n}(x)-2nH_{n-1}(x)\;\quad {\hbox{with}}\quad H_{0}(x)=1.$

The first few follow immediately from this relation,

$H_{1}=2x,\quad H_{2}=4x^{2}-2,\quad H_{3}=8x^{3}-12x,\quad \ldots$

Differential equation

The polynomials satisfy the Hermite differential equation for the special case that the coefficient of H_n(x) is equal to the even integer 2n,

${\frac {d^{2}H_{n}}{dx^{2}}}-2x\,{\frac {dH_{n}}{dx}}+2nH_{n}=0.$

Symmetry

$H_{n}(-x)=(-1)^{n}H_{n}(x)\;,$

the functions of even n are symmetric under x → −x and those of odd n are antisymmetric under this substitution.

Rodrigues' formula

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

Generating function

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

First few terms

$\left(1+2xt+{\frac {1}{2}}(2x)^{2}t^{2}+{\frac {1}{6}}(2x)^{3}t^{3}+\cdots \right)\left(1-t^{2}+\cdots \right)=1+2x\;t+(4x^{2}-2)\;{\frac {t^{2}}{2}}+(8x^{3}-12x)\;{\frac {t^{3}}{6}}+\cdots$

so that

$H_{0}=1,\quad H_{1}(x)=2x,\quad H_{2}(x)=4x^{2}-2,\quad H_{3}(x)=8x^{3}-12x.$

Differential relation

${\frac {dH_{n}(x)}{dx}}=2nH_{n-1}(x).$

Sum formula

$H_{n}(x+y)=\left({\frac {1}{\sqrt {2}}}\right)^{n}\sum _{k=0}^{n}{\binom {n}{k}}\;H_{k}({\sqrt {2}}\,x)\;H_{n-k}({\sqrt {2}}\,y),$

where ${\binom {n}{k}}$ is a binomial coefficient.

References

M. Abramowitz and I.A. Stegun (Eds), Handbook of Mathematical Functions, Dover, New York (1972). Chapter 22

Abramowitz and Stegun online

Eric W. Weisstein, Hermite Polynomial