1 #This file was created by <bruno> Sun Feb 16 14:24:48 1997
2 #LyX 0.10 (C) 1995 1996 Matthias Ettrich and the LyX Team
6 \catcode`@=11 % @ ist ab jetzt ein gewoehnlicher Buchstabe
7 \def\mod#1{\allowbreak \mkern8mu \mathop{\operator@font mod}\,\,{#1}}
8 \def\pmod#1{\allowbreak \mkern8mu \left({\mathop{\operator@font mod}\,\,{#1}}\right)}
9 \catcode`@=12 % @ ist ab jetzt wieder ein Sonderzeichen
21 \paragraph_separation indent
22 \quotes_language english
24 \paperorientation portrait
31 The Legendre polynomials
32 \begin_inset Formula \( P_{n}(x) \)
38 P_{n}(x)=\frac{1}{2^{n}n!}\cdot \left( \frac{d}{dx}\right) ^{n}(x^{2}-1)^{n}\]
42 (For a motivation of the
43 \begin_inset Formula \( 2^{n} \)
46 in the denominator, look at
47 \begin_inset Formula \( P_{n}(x) \)
51 \begin_inset Formula \( p \)
55 \begin_inset Formula \( P_{n}(x)\equiv P_{p-1-n}(x)\mod p \)
59 \begin_inset Formula \( 0\leq n\leq p-1 \)
63 This wouldn't hold if the
64 \begin_inset Formula \( 2^{n} \)
67 factor in the denominator weren't present.
75 \begin_inset Formula \( P_{n}(x) \)
78 satisfies the recurrence relation
94 (n+1)\cdot P_{n+1}(x)=(2n+1)x\cdot P_{n}(x)-n\cdot P_{n-1}(x)\]
99 \begin_inset Formula \( n\geq 0 \)
102 and the differential equation
103 \begin_inset Formula \( (1-x^{2})\cdot P_{n}^{''}(x)-2x\cdot P_{n}^{'}(x)+(n^{2}+n)\cdot P_{n}(x)=0 \)
107 \begin_inset Formula \( n\geq 0 \)
118 \begin_inset Formula \( F:=\sum ^{\infty }_{n=0}P_{n}(x)\cdot z^{n} \)
121 be the generating function of the sequence of polynomials.
122 It is the diagonal series of the power series
125 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}\]
129 Because the Taylor series development theorem holds in formal power series
130 rings (see [1], section 2.
134 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}\\
135 & = & \sum _{n=0}^{\infty }\frac{1}{2^{n}}\cdot \left( (x+y)^{2}-1\right) ^{n}\cdot z^{n}\\
136 & = & \frac{1}{1-\frac{1}{2}\left( (x+y)^{2}-1\right) z}
141 We take over the terminology from the
142 \begin_inset Quotes eld
146 \begin_inset Quotes erd
150 \begin_inset Formula \( R=Q[x] \)
154 \begin_inset Formula \( M=Q[[x]] \)
157 (or, if you like it better,
158 \begin_inset Formula \( M=H(C) \)
161 , the algebra of functions holomorphic in the entire complex plane).
163 \begin_inset Formula \( G\in M[[y,z]] \)
167 \begin_inset Formula \( F \)
171 \begin_inset Formula \( R[z] \)
175 Let's proceed exactly as in the
176 \begin_inset Quotes eld
180 \begin_inset Quotes erd
185 \begin_inset Formula \( F(z^{2}) \)
188 is the coefficient of
189 \begin_inset Formula \( t^{0} \)
195 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)}\]
199 The splitting field of the denominator is
200 \begin_inset Formula \( L=Q(x)(z)(\alpha ) \)
206 \alpha _{1/2}=\frac{1-xz^{2}\pm \sqrt{1-2xz^{2}+z^{4}}}{z^{3}}\]
213 \alpha =\alpha _{1}=\frac{2}{z^{3}}-\frac{2x}{z}+\frac{1-x^{2}}{2}z+\cdots \in Q(x)[[z]][\frac{1}{z}]\]
220 \alpha _{2}=\frac{x^{2}-1}{2}z+\cdots \in Q(x)[[z]][\frac{1}{z}]\]
224 The partial fraction decomposition of
225 \begin_inset Formula \( G(zt,\frac{z}{t}) \)
231 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) \]
238 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}}}\]
245 F(z)=\frac{1}{\sqrt{1-2xz+z^{2}}}\]
253 \begin_inset Formula \( (1-2xz+z^{2})\cdot \frac{d}{dz}F+(z-x)\cdot F=0 \)
257 This is equivalent to the claimed recurrence.
261 Starting from the closed form for
262 \begin_inset Formula \( F \)
265 , we compute a linear relation for the partial derivatives of
266 \begin_inset Formula \( F \)
271 \begin_inset Formula \( \partial _{x}=\frac{d}{dx} \)
275 \begin_inset Formula \( \Delta _{z}=z\frac{d}{dz} \)
289 \left( 1-2xz+z^{2}\right) \cdot \partial _{x}F=z\cdot F\]
296 \left( 1-2xz+z^{2}\right) ^{2}\cdot \partial _{x}^{2}F=3z^{2}\cdot F\]
303 \left( 1-2xz+z^{2}\right) \cdot \Delta _{z}F=(xz-z^{2})\cdot F\]
310 \left( 1-2xz+z^{2}\right) ^{2}\cdot \partial _{x}\Delta _{z}F=(z+xz^{2}-2z^{3})\cdot F\]
317 \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\]
322 \begin_inset Formula \( 5\times 6 \)
325 system of linear equations over
326 \begin_inset Formula \( Q(x) \)
332 \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\]
336 Divide by the first factor to get
339 (-2x)\cdot \partial _{x}F+(1-x^{2})\cdot \partial _{x}^{2}F+\Delta _{z}F+\Delta _{z}^{2}F=0\]
343 This is equivalent to the claimed equation
344 \begin_inset Formula \( (1-x^{2})\cdot P_{n}^{''}(x)-2x\cdot P_{n}^{'}(x)+(n^{2}+n)\cdot P_{n}(x)=0 \)
351 [1] Bruno Haible: D-finite power series in several variables.
354 Diploma thesis, University of Karlsruhe, June 1989