  This report addresses the moments, ${\mathfrak{G}_n}\left( {\mathbf{q}}
\right) = \int_{ - \infty }^{ + \infty } {{\omega ^n}S\left(
{{\mathbf{q}},\omega } \right)\mathrm{d}\omega },\,n \in \mathbb{N},\,n \geq -
1$, of the quantum mechanical dynamic structure factor $S\left(
{{\mathbf{q}},\omega } \right)$ for a one-component Coulomb plasma in
thermodynamic equilibrium. The Fluctuation Dissipation Theorem relates these
moments to integrals involving the imaginary part of the inverse longitudinal
dielectric function, with the odd moments in particular being equivalent to the
odd moments of the imaginary part of the inverse dielectric function.
Application of the Generalized Plasmon Pole Approximation arXiv:1508.05606
[physics.plasm-ph] to a weakly-coupled non-degenerate plasma, leads to general
formulae expressed in terms of polynomial functions. Explicit forms of these
functions are given for $n \leq 20$. These formulae are generalized to
degenerate and partially degenerate plasmas, in small-$\mathbf{q}$
(long-wavelength) regimes.
