doc/polynomial/hermite.tex

   1 %% This LaTeX-file was created by <bruno> Sun Feb 16 14:05:55 1997
   2 %% LyX 0.10 (C) 1995 1996 by Matthias Ettrich and the LyX Team
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 100 \begin{document}
 101
 102 The Hermite polynomials  \( H_{n}(x) \) are defined through
 103 \[
 104 H_{n}(x)=(-1)^{n}e^{x^{2}}\cdot \left( \frac{d}{dx}\right) ^{n}\left( e^{-x^{2}}\right) \]
 105
 106
 107 \begin{description}
 108
 109 \item [Theorem:]~
 110
 111 \end{description}
 112
 113  \( H_{n}(x) \) satisfies the recurrence relation
 114
 115
 116 \[
 117 H_{0}(x)=1\]
 118
 119
 120
 121 \[
 122 H_{n+1}(x)=2x\cdot H_{n}(x)-2n\cdot H_{n-1}(x)\]
 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  \( n\geq 0 \).
 124
 125 \begin{description}
 126
 127 \item [Proof:]~
 128
 129 \end{description}
 130
 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
 134 \begin{eqnarray*}
 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}}\\
 137  & = & e^{2xz-z^{2}}
 138 \end{eqnarray*}
 139 It follows
 140 that  \( \frac{d}{dz}F=(2x-2z)\cdot F \). This is equivalent to the claimed recurrence.
 141
 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
 144 \[
 145 F=1\cdot F\]
 146
 147 \[
 148 \partial _{x}F=2z\cdot F\]
 149
 150 \[
 151 \partial _{x}^{2}F=4z^{2}\cdot F\]
 152
 153 \[
 154 \Delta _{z}F=(2xz-2z^{2})\cdot F\]
 155
 156 \[
 157 \partial _{x}\Delta _{z}F=(2z+4xz^{2}-4z^{3})\cdot F\]
 158
 159 \[
 160 \Delta _{z}^{2}F=\left( 2x\cdot z+(4x^{2}-4)\cdot z^{2}-8x\cdot z^{3}+4\cdot z^{4}\right) \cdot F\]
 161  Solve
 162 a homogeneous  \( 5\times 6 \) system of linear equations over  \( Q(x) \) to get
 163 \[
 164 (-2x)\cdot \partial _{x}F+\partial _{x}^{2}F+2\cdot \Delta _{z}F=0\]
 165  This is
 166 equivalent to the claimed equation  \( H_{n}^{''}(x)-2x\cdot H_{n}^{'}(x)+2n\cdot H_{n}(x)=0 \).
 167
 168 \begin{lyxsectionbibliography}
 169
 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
 172 2.22.
 173
 174 \end{lyxsectionbibliography}
 175
 176 \end{document}