# Hermite equation

The linear differential equation

$$\frac{{d}^{2}f}{d{z}^{2}}-2z\frac{df}{dz}+2nf=\mathrm{\hspace{0.33em}0},$$ |

in which $n$ is a real , is called the Hermite equation. Its general solution is $f:=A{f}_{1}+B{f}_{2}$ with $A$ and $B$ arbitrary and the functions^{} ${f}_{1}$ and ${f}_{2}$ presented as

${f}_{1}(z):=z+\frac{2(1-n)}{3!}{z}^{3}+\frac{{2}^{2}(1-n)(3-n)}{5!}{z}^{5}+\frac{{2}^{3}(1-n)(3-n)(5-n)}{7!}{z}^{7}+\mathrm{\dots},$

${f}_{2}(z):=\mathrm{\hspace{0.33em}1}+\frac{2(-n)}{2!}{z}^{2}+\frac{{2}^{2}(-n)(2-n)}{4!}{z}^{4}+\frac{{2}^{3}(-n)(2-n)(4-n)}{6!}{z}^{6}+\mathrm{\dots}$

It’s easy to check that these power series^{} satisfy the differential equation. The coefficients ${b}_{\nu}$ in both series obey the recurrence

$${b}_{\nu}=\frac{2(\nu -2-n)}{\nu (nu-1)}{b}_{\nu -2}.$$ |

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

$$R=\underset{\nu \to \mathrm{\infty}}{lim}\left|\frac{{b}_{\nu -2}}{{b}_{\nu}}\right|=\underset{\nu \to \mathrm{\infty}}{lim}\frac{\nu}{2}\cdot \frac{1-1/\nu}{1-(n+2)/\nu}=\mathrm{\infty}.$$ |

Therefore the series converge in the whole complex plane^{} and define entire functions^{}.

If the $n$ is a non-negative integer, then one of ${f}_{1}$ and ${f}_{2}$ 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 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 |