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\textbf{Spectral Representation and Moments of a Green's function}
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Deriving the spectral representation 
\begin{equation}
\left\langle \left\{ c_{k}(t),c_{k}^{\dagger }(0)\right\} \right\rangle
=\sum_{nm}e^{-\beta \epsilon _{n}}\left\{ \left\langle n\left| c_{k}(t)\right|
m\right\rangle \left\langle m\left| c_{k}^{\dagger }(0)\right|
n\right\rangle +\left\langle n\left| c_{k}^{\dagger }(0)\right|
m\right\rangle \left\langle m\left| c_{k}(t)\right| n\right\rangle \right\}
\end{equation}
\begin{eqnarray*}
\int_{-\infty }^{\infty }dte^{i\omega t}\left\langle \left\{
c_{k}(t),c_{k}^{\dagger }(0)\right\} \right\rangle  &=&\sum_{nm}e^{-\beta
\epsilon _{n}}2\pi \delta (\omega -(\epsilon _{m}-\epsilon _{n}))\left|
\left\langle m\left| c_{k}^{\dagger }\right| n\right\rangle \right| ^{2} \\
&&+\sum_{nm}e^{-\beta \epsilon _{n}}2\pi \delta (\omega -(\epsilon
_{n}-\epsilon _{m}))\left| \left\langle m\left| c_{k}\right| n\right\rangle
\right| ^{2} \\
&=&2\pi A(k,\omega )
\end{eqnarray*}
Deriving the moments of the spectra using the equation of motion 
\begin{equation}
i\frac{\partial }{\partial t}c_{k}=[c_{k},H]
\end{equation}
\begin{eqnarray*}
2\pi A(k,\omega )\omega ^{p} &=&\int_{-\infty }^{\infty }\frac{1}{i^{p}}%
\left[ \frac{\partial ^{p}}{\partial t^{p}}e^{i\omega t}\right] \left\langle
\left\{ c_{k}(t),c_{k}^{\dagger }(0)\right\} \right\rangle dt \\
&=&\int_{-\infty }^{\infty }e^{i\omega t}\left\langle \left\{ \left[ \left[
c_{k},H\right] ,\stackrel{ptimes}{\cdot \cdot \cdot \cdot },H\right]
,c_{k}^{\dagger }(0)\right\} \right\rangle dt
\end{eqnarray*}
Therefore
\begin{equation}
\int_{-\infty }^{\infty }d\omega A(k,\omega )\omega ^{p}=\left\langle
\left\{ \left[ \left[ c_{k},H\right] ,\stackrel{ptimes}{\cdot \cdot \cdot
\cdot },H\right] ,c_{k}^{\dagger }(0)\right\} \right\rangle 
\end{equation}
and in particular
\begin{equation}
\int_{-\infty }^{\infty }d\omega A(k,\omega )\omega ^{p+q}=\left\langle
\left\{ \left[ \left[ c_{k},H\right] ,\stackrel{ptimes}{\cdot \cdot \cdot
\cdot },H\right] ,\left[ H,\stackrel{qtimes}{\cdot \cdot \cdot \cdot }\left[
H,c_{k}^{\dagger }\right] \right] \right\} \right\rangle 
\end{equation}

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