1 %% This LaTeX-file was created by <bruno> Sun Feb 16 14:24:52 1997
2 %% LyX 0.10 (C) 1995 1996 by Matthias Ettrich and the LyX Team
4 %% Don't edit this file unless you are sure what you are doing.
5 \documentclass[12pt,a4paper,oneside,onecolumn]{article}
7 \usepackage[latin1]{inputenc}
8 \usepackage[dvips]{epsfig}
11 %% BEGIN The lyx specific LaTeX commands.
15 \def\LyX{L\kern-.1667em\lower.25em\hbox{Y}\kern-.125emX\spacefactor1000}%
16 \newcommand{\lyxtitle}[1] {\thispagestyle{empty}
18 \section*{\LARGE \centering \sffamily \bfseries \protect#1 }
20 \newcommand{\lyxline}[1]{
21 {#1 \vspace{1ex} \hrule width \columnwidth \vspace{1ex}}
23 \newenvironment{lyxsectionbibliography}
26 \@mkboth{\uppercase{\refname}}{\uppercase{\refname}}
28 \itemindent-\leftmargin
30 \renewcommand{\makelabel}{}
34 \newenvironment{lyxchapterbibliography}
37 \@mkboth{\uppercase{\bibname}}{\uppercase{\bibname}}
39 \itemindent-\leftmargin
41 \renewcommand{\makelabel}{}
46 \newenvironment{lyxcode}
48 \rightmargin\leftmargin
57 \newcommand{\lyxlabel}[1]{#1 \hfill}
58 \newenvironment{lyxlist}[1]
60 {\settowidth{\labelwidth}{#1}
61 \setlength{\leftmargin}{\labelwidth}
62 \addtolength{\leftmargin}{\labelsep}
63 \renewcommand{\makelabel}{\lyxlabel}}}
65 \newcommand{\lyxletterstyle}{
66 \setlength\parskip{0.7em}
67 \setlength\parindent{0pt}
69 \newcommand{\lyxaddress}[1]{
74 \newcommand{\lyxrightaddress}[1]{
75 \par {\raggedleft \begin{tabular}{l}\ignorespaces
81 \newcommand{\lyxformula}[1]{
86 \newcommand{\lyxnumberedformula}[1]{
94 %% END The lyx specific LaTeX commands.
98 \setcounter{secnumdepth}{3}
99 \setcounter{tocdepth}{3}
101 %% Begin LyX user specified preamble:
102 \catcode`@=11 % @ ist ab jetzt ein gewoehnlicher Buchstabe
103 \def\mod#1{\allowbreak \mkern8mu \mathop{\operator@font mod}\,\,{#1}}
104 \def\pmod#1{\allowbreak \mkern8mu \left({\mathop{\operator@font mod}\,\,{#1}}\right)}
105 \catcode`@=12 % @ ist ab jetzt wieder ein Sonderzeichen
108 %% End LyX user specified preamble.
111 The Legendre polynomials \( P_{n}(x) \) are defined through
113 P_{n}(x)=\frac{1}{2^{n}n!}\cdot \left( \frac{d}{dx}\right) ^{n}(x^{2}-1)^{n}\]
115 of the \( 2^{n} \) in the denominator, look at \( P_{n}(x) \) modulo an odd prime \( p \), and
116 observe that \( P_{n}(x)\equiv P_{p-1-n}(x)\mod p \) for \( 0\leq n\leq p-1 \). This wouldn't hold if the \( 2^{n} \) factor in the denominator
125 \( P_{n}(x) \) satisfies the recurrence relation
134 (n+1)\cdot P_{n+1}(x)=(2n+1)x\cdot P_{n}(x)-n\cdot P_{n-1}(x)\]
135 for \( n\geq 0 \) and the differential equation \( (1-x^{2})\cdot P_{n}^{''}(x)-2x\cdot P_{n}^{'}(x)+(n^{2}+n)\cdot P_{n}(x)=0 \) for all \( n\geq 0 \).
143 Let \( F:=\sum ^{\infty }_{n=0}P_{n}(x)\cdot z^{n} \) be the generating function of the sequence of polynomials. It
144 is the diagonal series of the power series
146 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}\]
147 Because the Taylor series
148 development theorem holds in formal power series rings (see [1], section
149 2.16), we can simplify
151 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}\\
152 & = & \sum _{n=0}^{\infty }\frac{1}{2^{n}}\cdot \left( (x+y)^{2}-1\right) ^{n}\cdot z^{n}\\
153 & = & \frac{1}{1-\frac{1}{2}\left( (x+y)^{2}-1\right) z}
155 We take over the terminology from the ``diag\_rational''
156 paper; here \( R=Q[x] \) and \( M=Q[[x]] \) (or, if you like it better, \( M=H(C) \), the algebra of
157 functions holomorphic in the entire complex plane). \( G\in M[[y,z]] \) is rational;
158 hence \( F \) is algebraic over \( R[z] \). Let's proceed exactly as in the ``diag\_series''
159 paper. \( F(z^{2}) \) is the coefficient of \( t^{0} \) in
161 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)}\]
162 The splitting field of the denominator
163 is \( L=Q(x)(z)(\alpha ) \) where
165 \alpha _{1/2}=\frac{1-xz^{2}\pm \sqrt{1-2xz^{2}+z^{4}}}{z^{3}}\]
168 \alpha =\alpha _{1}=\frac{2}{z^{3}}-\frac{2x}{z}+\frac{1-x^{2}}{2}z+\cdots \in Q(x)[[z]][\frac{1}{z}]\]
171 \alpha _{2}=\frac{x^{2}-1}{2}z+\cdots \in Q(x)[[z]][\frac{1}{z}]\]
172 The partial fraction decomposition of \( G(zt,\frac{z}{t}) \) reads
174 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) \]
178 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}}}\]
181 F(z)=\frac{1}{\sqrt{1-2xz+z^{2}}}\]
184 It follows that \( (1-2xz+z^{2})\cdot \frac{d}{dz}F+(z-x)\cdot F=0 \). This is equivalent to the claimed recurrence.
186 Starting from the closed form for \( F \), we compute a linear relation
187 for the partial derivatives of \( F \). Write \( \partial _{x}=\frac{d}{dx} \) and \( \Delta _{z}=z\frac{d}{dz} \). One computes
192 \left( 1-2xz+z^{2}\right) \cdot \partial _{x}F=z\cdot F\]
195 \left( 1-2xz+z^{2}\right) ^{2}\cdot \partial _{x}^{2}F=3z^{2}\cdot F\]
198 \left( 1-2xz+z^{2}\right) \cdot \Delta _{z}F=(xz-z^{2})\cdot F\]
201 \left( 1-2xz+z^{2}\right) ^{2}\cdot \partial _{x}\Delta _{z}F=(z+xz^{2}-2z^{3})\cdot F\]
204 \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\]
206 a homogeneous \( 5\times 6 \) system of linear equations over \( Q(x) \) to get
208 \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\]
210 the first factor to get
212 (-2x)\cdot \partial _{x}F+(1-x^{2})\cdot \partial _{x}^{2}F+\Delta _{z}F+\Delta _{z}^{2}F=0\]
213 This is equivalent to the claimed equation
214 \( (1-x^{2})\cdot P_{n}^{''}(x)-2x\cdot P_{n}^{'}(x)+(n^{2}+n)\cdot P_{n}(x)=0 \).
216 \begin{lyxsectionbibliography}
218 \item [1] Bruno Haible: D-finite power series in several variables. \em Diploma
219 thesis, University of Karlsruhe, June 1989\em . Sections 2.14, 2.15
222 \end{lyxsectionbibliography}