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% definition.tex 
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% (c) 2022 Patrik Müller, Ostschweizer Fachhochschule
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\section{Definition
\label{laguerre:section:definition}}
\rhead{Definition}

\begin{align}
    x y''(x) + (1 - x) y'(x) + n y(x)
    =
    0 
    \label{laguerre:dgl}
\end{align}

\begin{align}
    L_n(x)
    =
    \sum_{k=0}^{n} 
    \frac{(-1)^k}{k!}
    \begin{pmatrix}
        n \\
        k
    \end{pmatrix}
    x^k
    \label{laguerre:polynom}
\end{align}

\begin{align}
    x y''(x) + (\alpha + 1 - x) y'(x) + n y(x)
    =
    0 
    \label{laguerre:generell_dgl}
\end{align}

\begin{align}
    L_n^\alpha (x)
    =
    \sum_{k=0}^{n} 
    \frac{(-1)^k}{k!}
    \begin{pmatrix}
        n + \alpha \\
        n - k
    \end{pmatrix}
    x^k
    \label{laguerre:polynom}
\end{align}