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.
The Hermite polynomials Hn(x) are orthogonal in the sense of the following inner product:
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 δn'n. The normalization constant is given by
Normalization is to unity
The polynomials NnHn(x) are orthonormal, which means that they are orthogonal and normalized to unity.
here if N even and if N odd.
Orthogonal polynomials can be constructed recursively by means of a Gram-Schmidt orthogonalization pocedure. This procedure yields the following relation
The first few follow immediately from this relation,
The polynomials satisfy the Hermite differential equation for the special case that the coefficient of Hn(x) is equal to the even integer 2n,
the functions of even n are symmetric under x → −x and those of odd n are antisymmetric under this substitution.
First few terms
where is a binomial coefficient.
M. Abramowitz and I.A. Stegun (Eds), Handbook of Mathematical Functions, Dover, New York (1972). Chapter 22