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361 lines
7.0 KiB
361 lines
7.0 KiB
#This file was created by <bruno> Sun Feb 16 14:24:48 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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\begin_preamble
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\catcode`@=11 % @ ist ab jetzt ein gewoehnlicher Buchstabe
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\def\mod#1{\allowbreak \mkern8mu \mathop{\operator@font mod}\,\,{#1}}
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\def\pmod#1{\allowbreak \mkern8mu \left({\mathop{\operator@font mod}\,\,{#1}}\right)}
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\catcode`@=12 % @ ist ab jetzt wieder ein Sonderzeichen
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\end_preamble
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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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\cursor 47
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The Legendre polynomials
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\begin_inset Formula \( P_{n}(x) \)
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\end_inset
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are defined through
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\begin_inset Formula
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\[
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P_{n}(x)=\frac{1}{2^{n}n!}\cdot \left( \frac{d}{dx}\right) ^{n}(x^{2}-1)^{n}\]
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\end_inset
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(For a motivation of the
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\begin_inset Formula \( 2^{n} \)
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\end_inset
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in the denominator, look at
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\begin_inset Formula \( P_{n}(x) \)
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\end_inset
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modulo an odd prime
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\begin_inset Formula \( p \)
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\end_inset
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, and observe that
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\begin_inset Formula \( P_{n}(x)\equiv P_{p-1-n}(x)\mod p \)
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\end_inset
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for
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\begin_inset Formula \( 0\leq n\leq p-1 \)
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\end_inset
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.
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This wouldn't hold if the
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\begin_inset Formula \( 2^{n} \)
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\end_inset
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factor in the denominator weren't present.
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)
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\layout Description
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Theorem:
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\layout Standard
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\begin_inset Formula \( P_{n}(x) \)
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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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\[
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P_{0}(x)=1\]
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\end_inset
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\layout Standard
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\begin_inset Formula
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\[
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(n+1)\cdot P_{n+1}(x)=(2n+1)x\cdot P_{n}(x)-n\cdot P_{n-1}(x)\]
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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 \( (1-x^{2})\cdot P_{n}^{''}(x)-2x\cdot P_{n}^{'}(x)+(n^{2}+n)\cdot P_{n}(x)=0 \)
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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 \( F:=\sum ^{\infty }_{n=0}P_{n}(x)\cdot z^{n} \)
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\end_inset
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be the generating function of the sequence of polynomials.
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It is the diagonal series of the power series
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\begin_inset Formula
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\[
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G:=\sum _{m,n=0}^{\infty }\frac{1}{2^{n}m!}\cdot \left( \frac{d}{dx}\right) ^{m}(x^{2}-1)^{n}\cdot y^{m}\cdot z^{n}\]
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\end_inset
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Because the Taylor series development theorem holds in formal power series
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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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G & = & \sum _{n=0}^{\infty }\frac{1}{2^{n}}\cdot \left( \sum _{m=0}^{\infty }\frac{1}{m!}\cdot \left( \frac{d}{dx}\right) ^{m}(x^{2}-1)^{n}\cdot y^{m}\right) \cdot z^{n}\\
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& = & \sum _{n=0}^{\infty }\frac{1}{2^{n}}\cdot \left( (x+y)^{2}-1\right) ^{n}\cdot z^{n}\\
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& = & \frac{1}{1-\frac{1}{2}\left( (x+y)^{2}-1\right) z}
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\end{eqnarray*}
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\end_inset
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We take over the terminology from the
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\begin_inset Quotes eld
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\end_inset
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diag_rational
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\begin_inset Quotes erd
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\end_inset
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paper; here
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\begin_inset Formula \( R=Q[x] \)
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\end_inset
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and
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\begin_inset Formula \( M=Q[[x]] \)
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\end_inset
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(or, if you like it better,
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\begin_inset Formula \( M=H(C) \)
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\end_inset
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, the algebra of functions holomorphic in the entire complex plane).
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\begin_inset Formula \( G\in M[[y,z]] \)
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\end_inset
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is rational; hence
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\begin_inset Formula \( F \)
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\end_inset
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is algebraic over
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\begin_inset Formula \( R[z] \)
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\end_inset
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.
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Let's proceed exactly as in the
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\begin_inset Quotes eld
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\end_inset
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diag_series
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\begin_inset Quotes erd
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\end_inset
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paper.
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\begin_inset Formula \( F(z^{2}) \)
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\end_inset
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is the coefficient of
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\begin_inset Formula \( t^{0} \)
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\end_inset
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in
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\begin_inset Formula
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\[
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G(zt,\frac{z}{t})=\frac{2t}{2t-\left( (x+zt)^{2}-1\right) z}=\frac{2t}{-z^{3}\cdot t^{2}+2(1-xz^{2})\cdot t+(z-x^{2}z)}\]
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\end_inset
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The splitting field of the denominator is
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\begin_inset Formula \( L=Q(x)(z)(\alpha ) \)
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\end_inset
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where
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\begin_inset Formula
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\[
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\alpha _{1/2}=\frac{1-xz^{2}\pm \sqrt{1-2xz^{2}+z^{4}}}{z^{3}}\]
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\end_inset
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\begin_inset Formula
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\[
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\alpha =\alpha _{1}=\frac{2}{z^{3}}-\frac{2x}{z}+\frac{1-x^{2}}{2}z+\cdots \in Q(x)[[z]][\frac{1}{z}]\]
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\end_inset
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\begin_inset Formula
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\[
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\alpha _{2}=\frac{x^{2}-1}{2}z+\cdots \in Q(x)[[z]][\frac{1}{z}]\]
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\end_inset
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The partial fraction decomposition of
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\begin_inset Formula \( G(zt,\frac{z}{t}) \)
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\end_inset
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reads
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\begin_inset Formula
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\[
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G(zt,\frac{z}{t})=-\frac{2}{z^{3}}\cdot \frac{1}{\alpha _{1}-\alpha _{2}}\cdot \left( \frac{\alpha _{1}}{t-\alpha _{1}}-\frac{\alpha _{2}}{t-\alpha _{2}}\right) \]
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\end_inset
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It follows that
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\begin_inset Formula
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\[
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F(z^{2})=-\frac{2}{z^{3}}\cdot \frac{1}{\alpha _{1}-\alpha _{2}}\cdot \left( \frac{\alpha _{1}}{0-\alpha _{1}}-0\right) =\frac{1}{\sqrt{1-2xz^{2}+z^{4}}}\]
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\end_inset
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hence
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\begin_inset Formula
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\[
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F(z)=\frac{1}{\sqrt{1-2xz+z^{2}}}\]
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\end_inset
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\layout Standard
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It follows that
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\begin_inset Formula \( (1-2xz+z^{2})\cdot \frac{d}{dz}F+(z-x)\cdot F=0 \)
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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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Starting from the closed form for
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\begin_inset Formula \( F \)
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\end_inset
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, we compute a linear relation for the partial 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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\[
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\left( 1-2xz+z^{2}\right) \cdot \partial _{x}F=z\cdot F\]
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\end_inset
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\begin_inset Formula
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\[
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\left( 1-2xz+z^{2}\right) ^{2}\cdot \partial _{x}^{2}F=3z^{2}\cdot F\]
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\end_inset
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\begin_inset Formula
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\[
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\left( 1-2xz+z^{2}\right) \cdot \Delta _{z}F=(xz-z^{2})\cdot F\]
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\end_inset
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\begin_inset Formula
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\[
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\left( 1-2xz+z^{2}\right) ^{2}\cdot \partial _{x}\Delta _{z}F=(z+xz^{2}-2z^{3})\cdot F\]
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\end_inset
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\begin_inset Formula
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\[
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\left( 1-2xz+z^{2}\right) ^{2}\cdot \Delta _{z}^{2}F=\left( xz+(x^{2}-2)z^{2}-xz^{3}+z^{4}\right) \cdot F\]
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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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\[
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\left( 1-2xz+z^{2}\right) ^{2}\cdot \left( (-2x)\cdot \partial _{x}F+(1-x^{2})\cdot \partial _{x}^{2}F+\Delta _{z}F+\Delta _{z}^{2}F\right) =0\]
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\end_inset
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Divide by the first factor to get
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\begin_inset Formula
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\[
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(-2x)\cdot \partial _{x}F+(1-x^{2})\cdot \partial _{x}^{2}F+\Delta _{z}F+\Delta _{z}^{2}F=0\]
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\end_inset
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This is equivalent to the claimed equation
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\begin_inset Formula \( (1-x^{2})\cdot P_{n}^{''}(x)-2x\cdot P_{n}^{'}(x)+(n^{2}+n)\cdot P_{n}(x)=0 \)
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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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14, 2.
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15 and 2.
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22.
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