Editing Classical orthogonal polynomials (section)

=== Hermite polynomials ===

The differential equation is

:<math>y'' - 2xy' + \lambda \,y = 0,\qquad \text{with}\qquad\lambda = 2n.</math>

This is '''Hermite's equation'''.

The second form of the differential equation is

:<math>(e^{-x^2}\,y')' + e^{-x^2}\,\lambda\,y = 0.</math>

The third form is

:<math>(e^{-x^2/2}\,y)'' + (\lambda +1-x^2)(e^{-x^2/2}\,y) = 0.</math>

The recurrence relation is

:<math>H_{n+1}(x) = 2x\,H_n(x) - 2n\,H_{n-1}(x).</math>

Rodrigues' formula is

:<math>H_n(x) = (-1)^n\,e^{x^2} \  \frac{d^n}{dx^n}\left(e^{-x^2}\right).</math>

The first few Hermite polynomials are

:<math>H_0(x) = 1</math>

:<math>H_1(x) = 2x</math>

:<math>H_2(x) = 4x^2-2</math>

:<math>H_3(x) = 8x^3-12x</math>

:<math>H_4(x) = 16x^4-48x^2+12</math>

One can define the '''associated Hermite functions'''

: <math> \psi_n(x) = (h_n)^{-1/2}\,e^{-x^2/2}H_n(x).</math>

Because the multiplier is proportional to the square root of the weight function, these functions
are orthogonal over <math>(-\infty, \infty)</math> with no weight function.

The third form of the differential equation above, for the associated Hermite functions, is

:<math>\psi'' + (\lambda +1-x^2)\psi = 0.</math>

The associated Hermite functions arise in many areas of mathematics and physics.
In quantum mechanics, they are the solutions of Schrödinger's equation for the harmonic oscillator.
They are also eigenfunctions (with eigenvalue (&minus;''i'' <sup>''n''</sup>) of the [[continuous Fourier transform]].

Many authors, particularly probabilists, use an alternate definition of the Hermite polynomials, with a weight function of <math>e^{-x^2/2}</math> instead of <math>e^{-x^2}</math>.  If the notation ''He'' is used for these Hermite polynomials, and ''H'' for those above, then these may be characterized by

:<math>He_n(x) = 2^{-n/2}\,H_n\left(\frac{x}{\sqrt{2}}\right).</math>

For further details, see [[Hermite polynomials]].