# Hermite polynomial

*13 January 2011*.

## Contents |

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.

## [edit] Orthonormality

The Hermite polynomials *H*_{n}(*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(−*x*²). Their orthogonality is expressed by the appearance of the Kronecker delta
δ_{n'n}. The normalization constant is given by

Normalization is to unity

The polynomials *N*_{n}*H*_{n}(*x*) are *orthonormal*, which means that they are orthogonal and normalized to unity.

## [edit] Explicit expression

here if *N* even and if *N* odd.

## [edit] Recursion relation

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

The first few follow immediately from this relation,

## [edit] Differential equation

The polynomials *H*_{n}(*x*) satisfy the Hermite differential equation

for the special case , i.e., for natural positive α.

- If α ≡
*n*is even, the solution is a polynomial of order*n*consisting of even powers of*x*only. - If α ≡
*n*is odd, the solution is a polynomial of order*n*consisting of odd powers of*x*only.

## [edit] Symmetry

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

## [edit] Rodrigues' formula

## [edit] Generating function

First few terms

so that

## [edit] Differential relation

## [edit] Sum formula

where is a binomial coefficient.

## [edit] References

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

Eric W. Weisstein, Hermite Polynomial