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#This file was created by <bruno> Sun Feb 16 00:38:14 1997#LyX 0.10 (C) 1995 1996 Matthias Ettrich and the LyX Team\lyxformat 2.10\textclass article\language default\inputencoding latin1\fontscheme default\epsfig dvips\papersize a4paper \paperfontsize 12 \baselinestretch 1.00 \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \quotes_language english \quotes_times 2 \paperorientation portrait \papercolumns 0 \papersides 1 \paperpagestyle plain
\layout Standard
The Hermite polynomials \begin_inset Formula \( H_{n}(x) \)\end_inset
are defined through \begin_inset Formula \[H_{n}(x)=(-1)^{n}e^{x^{2}}\cdot \left( \frac{d}{dx}\right) ^{n}\left( e^{-x^{2}}\right) \]
\end_inset
\layout Description
Theorem:\layout Standard
\begin_inset Formula \( H_{n}(x) \)\end_inset
satisfies the recurrence relation\layout Standard
\begin_inset Formula \[H_{0}(x)=1\]
\end_inset
\layout Standard
\begin_inset Formula \[H_{n+1}(x)=2x\cdot H_{n}(x)-2n\cdot H_{n-1}(x)\]
\end_inset
for \begin_inset Formula \( n\geq 0 \)\end_inset
and the differential equation \begin_inset Formula \( H_{n}^{''}(x)-2x\cdot H_{n}^{'}(x)+2n\cdot H_{n}(x)=0 \)\end_inset
for all \begin_inset Formula \( n\geq 0 \)\end_inset
.
\layout Description
Proof:\layout Standard
Let \begin_inset Formula \( F:=\sum ^{\infty }_{n=0}\frac{H_{n}(x)}{n!}z^{n} \)\end_inset
be the exponential generating function of the sequence of polynomials. Then, because the Taylor series development theorem holds in formal power series rings (see [1], section 2.16), we can simplify\begin_inset Formula \begin{eqnarray*}F & = & e^{x^{2}}\cdot \sum ^{\infty }_{n=0}\frac{1}{n!}\left( \frac{d}{dx}\right) ^{n}\left( e^{-x^{2}}\right) \cdot (-z)^{n}\\ & = & e^{x^{2}}\cdot e^{-(x-z)^{2}}\\ & = & e^{2xz-z^{2}}\end{eqnarray*}
\end_inset
It follows that \begin_inset Formula \( \frac{d}{dz}F=(2x-2z)\cdot F \)\end_inset
. This is equivalent to the claimed recurrence.
\layout Standard\cursor 190 Starting from this equation, we compute a linear relation for the partial derivatives of \begin_inset Formula \( F \)\end_inset
. Write \begin_inset Formula \( \partial _{x}=\frac{d}{dx} \)\end_inset
and \begin_inset Formula \( \Delta _{z}=z\frac{d}{dz} \)\end_inset
. One computes\begin_inset Formula \[F=1\cdot F\]
\end_inset
\begin_inset Formula \[\partial _{x}F=2z\cdot F\]
\end_inset
\begin_inset Formula \[\partial _{x}^{2}F=4z^{2}\cdot F\]
\end_inset
\begin_inset Formula \[\Delta _{z}F=(2xz-2z^{2})\cdot F\]
\end_inset
\begin_inset Formula \[\partial _{x}\Delta _{z}F=(2z+4xz^{2}-4z^{3})\cdot F\]
\end_inset
\begin_inset Formula \[\Delta _{z}^{2}F=\left( 2x\cdot z+(4x^{2}-4)\cdot z^{2}-8x\cdot z^{3}+4\cdot z^{4}\right) \cdot F\]
\end_inset
Solve a homogeneous \begin_inset Formula \( 5\times 6 \)\end_inset
system of linear equations over \begin_inset Formula \( Q(x) \)\end_inset
to get \begin_inset Formula \[(-2x)\cdot \partial _{x}F+\partial _{x}^{2}F+2\cdot \Delta _{z}F=0\]
\end_inset
This is equivalent to the claimed equation \begin_inset Formula \( H_{n}^{''}(x)-2x\cdot H_{n}^{'}(x)+2n\cdot H_{n}(x)=0 \)\end_inset
.
\layout Bibliography
[1] Bruno Haible: D-finite power series in several variables. \shape italic Diploma thesis, University of Karlsruhe, June 1989\shape default . Sections 2.15 and 2.22.
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