Hermite equation

The linear differential equation

d2fdz2-2zdfdz+2nf= 0,

in which n is a real , is called the Hermite equation.  Its general solution is  f:=Af1+Bf2  with A and B arbitrary and the functionsMathworldPlanetmath f1 and f2 presented as


f2(z):= 1+2(-n)2!z2+22(-n)(2-n)4!z4+23(-n)(2-n)(4-n)6!z6+

It’s easy to check that these power seriesMathworldPlanetmath satisfy the differential equation.  The coefficients bν in both series obey the recurrence


Thus we have the radii of convergence (http://planetmath.org/RadiusOfConvergence)


Therefore the series converge in the whole complex planeMathworldPlanetmath and define entire functionsMathworldPlanetmath.

If the n is a non-negative integer, then one of f1 and f2 is simply a polynomial function.  The polynomial solutions of the Hermite equation are usually normed so that the highest degree (http://planetmath.org/PolynomialRing) is (2z)n and called the Hermite polynomialsDlmfDlmfDlmfMathworldPlanetmath.

Title Hermite equation
Canonical name HermiteEquation
Date of creation 2013-03-22 15:16:15
Last modified on 2013-03-22 15:16:15
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 19
Author pahio (2872)
Entry type Definition
Classification msc 34M05
Synonym Hermite differential equation
Related topic ChebyshevEquation