  We consider the two-dimensional one-component plasma (jellium) of mobile
pointlike particles with the same charge $e$, interacting pairwisely by the
logarithmic Coulomb potential and immersed in a fixed neutralizing background
charge density. Particles are in thermal equilibrium at the inverse temperature
$\beta$, the only relevant dimensionless parameter is the coupling constant
$\Gamma\equiv\beta e^2$. In the bulk fluid regime and for any value of the
coupling constant $\Gamma=2\times{\rm integer}$, \v{S}amaj and Percus [J. Stat.
Phys. {\bf 80}, 811--824 (1995)] have derived an infinite sequence of sum rules
for the coefficients of the short-distance expansion of particle pair
correlation function. In the context of the equivalent fractional quantum Hall
effect, by using specific methods of quantum geometry Haldane [PRL {\bf 107},
116801 (2011) and arXiv:1112.0990v2] derived a self-dual relation for the
Landau-level guiding-center structure factor. In this paper, we establish the
relation between the guiding-center structure factor and the pair correlation
function of jellium particles. It is shown that the self-dual formula, which
provides an exact relation between the pair correlation function and its
Fourier component, comes directly from the short-distance symmetry of the bulk
jellium. The short-distant symmetry of pair correlations is extended to the
semi-infinite geometry of a rectilinear plain hard wall with a fixed surface
charge density, constraining particles to a half-space. The symmetry is derived
for the original jellium model as well as its simplified version with no
background charge (charged wall surface with ``counter-ions only''). The
obtained results are checked at the exactly solvable free-fermion coupling
$\Gamma=2$.
