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111 The Laguerre polynomials \( L_{n}(x) \) are defined through
113 L_{n}(x)=e^{x}\cdot \left( \frac{d}{dx}\right) ^{n}(x^{n}e^{-x})\]
122 \( L_{n}(x) \) satisfies the recurrence relation
131 L_{n+1}(x)=(2n+1-x)\cdot L_{n}(x)-n^{2}\cdot L_{n-1}(x)\]
132 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 \( n\geq 0 \).
140 Let \( F:=\sum ^{\infty }_{n=0}\frac{L_{n}(x)}{n!}\cdot z^{n} \) be the exponential generating function of the sequence of polynomials.
141 It is the diagonal series of the power series
143 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}\]
144 Because the Taylor series
145 development theorem holds in formal power series rings (see [1], section
146 2.16), we can simplify
148 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}\\
149 & = & e^{x}\cdot \sum _{n=0}^{\infty }(x+y)^{n}e^{-(x+y)}\cdot z^{n}\\
150 & = & \frac{e^{-y}}{1-(x+y)z}
152 We take over the terminology from the ``diag\_rational''
153 paper; here \( R=Q[x] \) and \( M=Q[[x]] \) (or, if you like it better, \( M=H(C) \), the algebra of
154 functions holomorphic in the entire complex plane). \( G\in M[[y,z]] \) is not rational;
155 nevertheless we can proceed similarly to the ``diag\_series'' paper.
156 \( F(z^{2}) \) is the coefficient of \( t^{0} \) in
158 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 \]
159 The denominator's only zero is \( t=\frac{xz}{1-z^{2}} \). We
162 e^{-zt}=e^{-\frac{xz^{2}}{1-z^{2}}}+\left( zt-\frac{xz^{2}}{1-z^{2}}\right) \cdot P(z,t)\]
163 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 \( M\ll z,t\gg \)
166 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)\\
167 & = & \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)
169 Here, the coefficient of \( t^{0} \) is
171 F(z^{2})=\frac{1}{1-z^{2}}\cdot e^{-\frac{xz^{2}}{1-z^{2}}}\]
174 F(z)=\frac{1}{1-z}\cdot e^{-\frac{xz}{1-z}}\]
177 It follows that \( (1-z)^{2}\cdot \frac{d}{dz}F-(1-x-z)\cdot F=0 \). This is equivalent to the claimed recurrence.
179 Starting from the closed form for \( F \), we compute a linear relation
180 for the partial derivatives of \( F \). Write \( \partial _{x}=\frac{d}{dx} \) and \( \Delta _{z}=z\frac{d}{dz} \). One computes
185 \left( 1-z\right) \cdot \partial _{x}F=-z\cdot F\]
188 \left( 1-z\right) ^{2}\cdot \partial _{x}^{2}F=z^{2}\cdot F\]
191 \left( 1-z\right) ^{2}\cdot \Delta _{z}F=((1-x)z-z^{2})\cdot F\]
194 \left( 1-z\right) ^{3}\cdot \partial _{x}\Delta _{z}F=(-z+xz^{2}+z^{3})\cdot F\]
196 a homogeneous \( 4\times 5 \) system of linear equations over \( Q(x) \) to get
198 \left( 1-z\right) ^{3}\cdot \left( (1-x)\cdot \partial _{x}F+x\cdot \partial _{x}^{2}F+\Delta _{z}F\right) =0\]
200 the first factor to get
202 (1-x)\cdot \partial _{x}F+x\cdot \partial _{x}^{2}F+\Delta _{z}F=0\]
203 This is equivalent to the claimed equation
204 \( x\cdot L_{n}^{''}(x)+(1-x)\cdot L_{n}^{'}(x)+n\cdot L_{n}(x)=0 \).
206 \begin{lyxsectionbibliography}
208 \item [1] Bruno Haible: D-finite power series in several variables. \em Diploma
209 thesis, University of Karlsruhe, June 1989\em . Sections 2.15 and
212 \end{lyxsectionbibliography}