1 %% This LaTeX-file was created by <bruno> Sun Feb 16 14:05:55 1997
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102 The Hermite polynomials \( H_{n}(x) \) are defined through
104 H_{n}(x)=(-1)^{n}e^{x^{2}}\cdot \left( \frac{d}{dx}\right) ^{n}\left( e^{-x^{2}}\right) \]
113 \( H_{n}(x) \) satisfies the recurrence relation
122 H_{n+1}(x)=2x\cdot H_{n}(x)-2n\cdot H_{n-1}(x)\]
123 for \( n\geq 0 \) and the differential equation \( H_{n}^{''}(x)-2x\cdot H_{n}^{'}(x)+2n\cdot H_{n}(x)=0 \) for all \( n\geq 0 \).
131 Let \( F:=\sum ^{\infty }_{n=0}\frac{H_{n}(x)}{n!}z^{n} \) be the exponential generating function of the sequence of polynomials.
132 Then, because the Taylor series development theorem holds in formal
133 power series rings (see [1], section 2.16), we can simplify
135 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}\\
136 & = & e^{x^{2}}\cdot e^{-(x-z)^{2}}\\
140 that \( \frac{d}{dz}F=(2x-2z)\cdot F \). This is equivalent to the claimed recurrence.
142 Starting from this equation, we compute a linear relation for the
143 partial derivatives of \( F \). Write \( \partial _{x}=\frac{d}{dx} \) and \( \Delta _{z}=z\frac{d}{dz} \). One computes
148 \partial _{x}F=2z\cdot F\]
151 \partial _{x}^{2}F=4z^{2}\cdot F\]
154 \Delta _{z}F=(2xz-2z^{2})\cdot F\]
157 \partial _{x}\Delta _{z}F=(2z+4xz^{2}-4z^{3})\cdot F\]
160 \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\]
162 a homogeneous \( 5\times 6 \) system of linear equations over \( Q(x) \) to get
164 (-2x)\cdot \partial _{x}F+\partial _{x}^{2}F+2\cdot \Delta _{z}F=0\]
166 equivalent to the claimed equation \( H_{n}^{''}(x)-2x\cdot H_{n}^{'}(x)+2n\cdot H_{n}(x)=0 \).
168 \begin{lyxsectionbibliography}
170 \item [1] Bruno Haible: D-finite power series in several variables. \em Diploma
171 thesis, University of Karlsruhe, June 1989\em . Sections 2.15 and
174 \end{lyxsectionbibliography}