Hermite polynomial

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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 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.

Explicit expression

here if N even and 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

The first few follow immediately from this relation,

Differential equation

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

Symmetry

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

Rodrigues' formula

Generating function

First few terms

so that

Differential relation

Sum formula

where 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