Hermite equation
The linear differential equation
in which is a real , is called the Hermite equation. Its general solution is with and arbitrary and the functions and presented as
It’s easy to check that these power series satisfy the differential equation. The coefficients 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 plane and define entire functions.
If the is a non-negative integer, then one of and 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 and called the Hermite polynomials.
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 |