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\begin{document}

The Hermite polynomials  \( H_{n}(x) \) are defined through 
\[
H_{n}(x)=(-1)^{n}e^{x^{2}}\cdot \left( \frac{d}{dx}\right) ^{n}\left( e^{-x^{2}}\right) \]


\begin{description}

\item [Theorem:]~

\end{description}

 \( H_{n}(x) \) satisfies the recurrence relation


\[
H_{0}(x)=1\]



\[
H_{n+1}(x)=2x\cdot H_{n}(x)-2n\cdot H_{n-1}(x)\]
 for  \( n\geq 0 \) and the differential equation  \( H_{n}^{''}(x)-2x\cdot H_{n}^{'}(x)+2n\cdot H_{n}(x)=0 \) for all  \( n\geq 0 \).

\begin{description}

\item [Proof:]~

\end{description}

Let  \( F:=\sum ^{\infty }_{n=0}\frac{H_{n}(x)}{n!}z^{n} \) 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{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*}
It follows
that  \( \frac{d}{dz}F=(2x-2z)\cdot F \). This is equivalent to the claimed recurrence.

Starting from this equation, we compute a linear relation for the
partial derivatives of  \( F \). Write  \( \partial _{x}=\frac{d}{dx} \) and  \( \Delta _{z}=z\frac{d}{dz} \). One computes
\[
F=1\cdot F\]

\[
\partial _{x}F=2z\cdot F\]

\[
\partial _{x}^{2}F=4z^{2}\cdot F\]

\[
\Delta _{z}F=(2xz-2z^{2})\cdot F\]

\[
\partial _{x}\Delta _{z}F=(2z+4xz^{2}-4z^{3})\cdot F\]

\[
\Delta _{z}^{2}F=\left( 2x\cdot z+(4x^{2}-4)\cdot z^{2}-8x\cdot z^{3}+4\cdot z^{4}\right) \cdot F\]
 Solve
a homogeneous  \( 5\times 6 \) system of linear equations over  \( Q(x) \) to get 
\[
(-2x)\cdot \partial _{x}F+\partial _{x}^{2}F+2\cdot \Delta _{z}F=0\]
 This is
equivalent to the claimed equation  \( H_{n}^{''}(x)-2x\cdot H_{n}^{'}(x)+2n\cdot H_{n}(x)=0 \).

\begin{lyxsectionbibliography}

\item [1] Bruno Haible: D-finite power series in several variables. \em Diploma
thesis, University of Karlsruhe, June 1989\em . Sections 2.15 and
2.22.

\end{lyxsectionbibliography}

\end{document}
