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\begin{document}
The Laguerre polynomials \( L_{n}(x) \) are defined through \[
L_{n}(x)=e^{x}\cdot \left( \frac{d}{dx}\right) ^{n}(x^{n}e^{-x})\]
\begin{description}
\item [Theorem:]~
\end{description}
\( L_{n}(x) \) satisfies the recurrence relation
\[
L_{0}(x)=1\]
\[
L_{n+1}(x)=(2n+1-x)\cdot L_{n}(x)-n^{2}\cdot L_{n-1}(x)\]for \( n\geq 0 \) and the differential equation \( x\cdot L_{n}^{''}(x)+(1-x)\cdot L_{n}^{'}(x)+n\cdot L_{n}(x)=0 \) for all \( n\geq 0 \).
\begin{description}
\item [Proof:]~
\end{description}
Let \( F:=\sum ^{\infty }_{n=0}\frac{L_{n}(x)}{n!}\cdot z^{n} \) be the exponential generating function of the sequence of polynomials.It is the diagonal series of the power series\[
G:=\sum _{m,n=0}^{\infty }\frac{1}{m!}\cdot e^{x}\cdot \left( \frac{d}{dx}\right) ^{m}(x^{n}e^{-x})\cdot y^{m}\cdot z^{n}\]Because the Taylor seriesdevelopment theorem holds in formal power series rings (see [1], section2.16), we can simplify\begin{eqnarray*}G & = & e^{x}\cdot \sum _{n=0}^{\infty }\left( \sum _{m=0}^{\infty }\frac{1}{m!}\cdot \left( \frac{d}{dx}\right) ^{m}(x^{n}e^{-x})\cdot y^{m}\right) \cdot z^{n}\\ & = & e^{x}\cdot \sum _{n=0}^{\infty }(x+y)^{n}e^{-(x+y)}\cdot z^{n}\\ & = & \frac{e^{-y}}{1-(x+y)z}\end{eqnarray*}We take over the terminology from the ``diag\_rational''paper; here \( R=Q[x] \) and \( M=Q[[x]] \) (or, if you like it better, \( M=H(C) \), the algebra offunctions holomorphic in the entire complex plane). \( G\in M[[y,z]] \) is not rational;nevertheless we can proceed similarly to the ``diag\_series'' paper. \( F(z^{2}) \) is the coefficient of \( t^{0} \) in\[
G(zt,\frac{z}{t})=\frac{e^{-zt}}{1-z^{2}-\frac{xz}{t}}\in M[[zt,\frac{z}{t},z]]=M\ll z,t\gg \]The denominator's only zero is \( t=\frac{xz}{1-z^{2}} \). Wecan write\[
e^{-zt}=e^{-\frac{xz^{2}}{1-z^{2}}}+\left( zt-\frac{xz^{2}}{1-z^{2}}\right) \cdot P(z,t)\]with \( P(z,t)\in Q[[zt,\frac{xz^{2}}{1-z^{2}}]]\subset Q[[zt,x,z]]=M[[zt,z]]\subset M\ll z,t\gg \). This yields -- all computations being done in \( M\ll z,t\gg \)--\begin{eqnarray*}G(zt,\frac{z}{t}) & = & \frac{e^{-\frac{xz^{2}}{1-z^{2}}}}{1-z^{2}-\frac{xz}{t}}+\frac{zt}{1-z^{2}}\cdot P(z,t)\\ & = & \frac{1}{1-z^{2}}\cdot e^{-\frac{xz^{2}}{1-z^{2}}}\cdot \sum _{j=0}^{\infty }\left( \frac{x}{1-z^{2}}\frac{z}{t}\right) ^{j}+\frac{zt}{1-z^{2}}\cdot P(z,t)\end{eqnarray*}Here, the coefficient of \( t^{0} \) is\[
F(z^{2})=\frac{1}{1-z^{2}}\cdot e^{-\frac{xz^{2}}{1-z^{2}}}\]hence\[
F(z)=\frac{1}{1-z}\cdot e^{-\frac{xz}{1-z}}\]
It follows that \( (1-z)^{2}\cdot \frac{d}{dz}F-(1-x-z)\cdot F=0 \). This is equivalent to the claimed recurrence.
Starting from the closed form for \( F \), we compute a linear relationfor the partial derivatives of \( F \). Write \( \partial _{x}=\frac{d}{dx} \) and \( \Delta _{z}=z\frac{d}{dz} \). One computes\[
F=1\cdot F\]
\[
\left( 1-z\right) \cdot \partial _{x}F=-z\cdot F\]
\[
\left( 1-z\right) ^{2}\cdot \partial _{x}^{2}F=z^{2}\cdot F\]
\[
\left( 1-z\right) ^{2}\cdot \Delta _{z}F=((1-x)z-z^{2})\cdot F\]
\[
\left( 1-z\right) ^{3}\cdot \partial _{x}\Delta _{z}F=(-z+xz^{2}+z^{3})\cdot F\]Solvea homogeneous \( 4\times 5 \) system of linear equations over \( Q(x) \) to get \[
\left( 1-z\right) ^{3}\cdot \left( (1-x)\cdot \partial _{x}F+x\cdot \partial _{x}^{2}F+\Delta _{z}F\right) =0\]Divide bythe first factor to get\[
(1-x)\cdot \partial _{x}F+x\cdot \partial _{x}^{2}F+\Delta _{z}F=0\]This is equivalent to the claimed equation \( x\cdot L_{n}^{''}(x)+(1-x)\cdot L_{n}^{'}(x)+n\cdot L_{n}(x)=0 \).
\begin{lyxsectionbibliography}
\item [1] Bruno Haible: D-finite power series in several variables. \em Diplomathesis, University of Karlsruhe, June 1989\em . Sections 2.15 and2.22.
\end{lyxsectionbibliography}
\end{document}
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