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199 lines
3.4 KiB
199 lines
3.4 KiB
#This file was created by <bruno> Sun Feb 16 00:38:14 1997
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#LyX 0.10 (C) 1995 1996 Matthias Ettrich and the LyX Team
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\lyxformat 2.10
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\textclass article
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\language default
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\inputencoding latin1
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\fontscheme default
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\epsfig dvips
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\papersize a4paper
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\paperfontsize 12
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\baselinestretch 1.00
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\secnumdepth 3
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\tocdepth 3
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\paragraph_separation indent
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\quotes_language english
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\quotes_times 2
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\paperorientation portrait
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\papercolumns 0
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\papersides 1
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\paperpagestyle plain
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\layout Standard
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The Hermite polynomials
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\begin_inset Formula \( H_{n}(x) \)
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\end_inset
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are defined through
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\begin_inset Formula
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\[
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H_{n}(x)=(-1)^{n}e^{x^{2}}\cdot \left( \frac{d}{dx}\right) ^{n}\left( e^{-x^{2}}\right) \]
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\end_inset
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\layout Description
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Theorem:
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\layout Standard
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\begin_inset Formula \( H_{n}(x) \)
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\end_inset
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satisfies the recurrence relation
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\layout Standard
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\begin_inset Formula
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\[
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H_{0}(x)=1\]
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\end_inset
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\layout Standard
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\begin_inset Formula
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\[
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H_{n+1}(x)=2x\cdot H_{n}(x)-2n\cdot H_{n-1}(x)\]
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\end_inset
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for
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\begin_inset Formula \( n\geq 0 \)
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\end_inset
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and the differential equation
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\begin_inset Formula \( H_{n}^{''}(x)-2x\cdot H_{n}^{'}(x)+2n\cdot H_{n}(x)=0 \)
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\end_inset
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for all
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\begin_inset Formula \( n\geq 0 \)
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\end_inset
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.
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\layout Description
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Proof:
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\layout Standard
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Let
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\begin_inset Formula \( F:=\sum ^{\infty }_{n=0}\frac{H_{n}(x)}{n!}z^{n} \)
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\end_inset
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be the exponential generating function of the sequence of polynomials.
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Then, because the Taylor series development theorem holds in formal power
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series rings (see [1], section 2.
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16), we can simplify
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\begin_inset Formula
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\begin{eqnarray*}
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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}\\
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& = & e^{x^{2}}\cdot e^{-(x-z)^{2}}\\
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& = & e^{2xz-z^{2}}
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\end{eqnarray*}
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\end_inset
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It follows that
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\begin_inset Formula \( \frac{d}{dz}F=(2x-2z)\cdot F \)
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\end_inset
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.
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This is equivalent to the claimed recurrence.
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\layout Standard
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\cursor 190
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Starting from this equation, we compute a linear relation for the partial
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derivatives of
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\begin_inset Formula \( F \)
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\end_inset
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.
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Write
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\begin_inset Formula \( \partial _{x}=\frac{d}{dx} \)
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\end_inset
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and
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\begin_inset Formula \( \Delta _{z}=z\frac{d}{dz} \)
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\end_inset
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.
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One computes
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\begin_inset Formula
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\[
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F=1\cdot F\]
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\end_inset
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\begin_inset Formula
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\[
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\partial _{x}F=2z\cdot F\]
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\end_inset
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\begin_inset Formula
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\[
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\partial _{x}^{2}F=4z^{2}\cdot F\]
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\end_inset
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\begin_inset Formula
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\[
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\Delta _{z}F=(2xz-2z^{2})\cdot F\]
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\end_inset
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\begin_inset Formula
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\[
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\partial _{x}\Delta _{z}F=(2z+4xz^{2}-4z^{3})\cdot F\]
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\end_inset
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\begin_inset Formula
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\[
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\Delta _{z}^{2}F=\left( 2x\cdot z+(4x^{2}-4)\cdot z^{2}-8x\cdot z^{3}+4\cdot z^{4}\right) \cdot F\]
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\end_inset
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Solve a homogeneous
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\begin_inset Formula \( 5\times 6 \)
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\end_inset
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system of linear equations over
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\begin_inset Formula \( Q(x) \)
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\end_inset
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to get
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\begin_inset Formula
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\[
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(-2x)\cdot \partial _{x}F+\partial _{x}^{2}F+2\cdot \Delta _{z}F=0\]
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\end_inset
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This is equivalent to the claimed equation
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\begin_inset Formula \( H_{n}^{''}(x)-2x\cdot H_{n}^{'}(x)+2n\cdot H_{n}(x)=0 \)
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\end_inset
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.
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\layout Bibliography
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[1] Bruno Haible: D-finite power series in several variables.
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\shape italic
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Diploma thesis, University of Karlsruhe, June 1989
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\shape default
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.
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Sections 2.
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15 and 2.
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22.
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