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# Hermite equation

The linear differential equation

$\frac{d^{2}f}{dz^{2}}-2z\frac{df}{dz}+2nf\;=\;0,$ |

in which $n$ is a real constant, is called the Hermite equation. Its general solution is $f:=Af_{1}\!+\!Bf_{2}$ with $A$ and $B$ arbitrary constants and the functions $f_{1}$ and $f_{2}$ presented as

$f_{1}(z)\;:=\;z+\frac{2(1-n)}{3!}z^{3}+\frac{2^{2}(1-n)(3-n)}{5!}z^{5}+\frac{2%
^{3}(1-n)(3-n)(5-n)}{7!}z^{7}+\ldots\!,$

$f_{2}(z)\;:=\;1+\frac{2(-n)}{2!}z^{2}+\frac{2^{2}(-n)(2-n)}{4!}z^{4}+\frac{2^{%
3}(-n)(2-n)(4-n)}{6!}z^{6}+\ldots$

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

$b_{\nu}\;=\;\frac{2(\nu\!-\!2\!-\!n)}{\nu(nu\!-\!1)}b_{{\nu\!-\!2}}.$ |

Thus we have the radii of convergence

$R\;=\;\lim_{{\nu\to\infty}}\left|\frac{b_{{\nu-2}}}{b_{\nu}}\right|\;=\;\lim_{% {\nu\to\infty}}\frac{\nu}{2}\!\cdot\!\frac{1\!-\!1/\nu}{1\!-\!(n\!+\!2)/\nu}\;% =\;\infty.$ |

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

If the constant $n$ 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 term is $(2z)^{n}$ and called the Hermite polynomials.

## Mathematics Subject Classification

34M05*no label found*

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