  The structural properties of single component fluids of hard hyperspheres in
odd space dimensionalities $d$ are studied with an analytical approximation
method that generalizes the Rational Function Approximation earlier introduced
in the study of hard-sphere fluids [S. B. Yuste and A. Santos, Phys. Rev. A
{\bf 43}, 5418 (1991)]. The theory makes use of the exact form of the radial
distribution function to first order in density and extends it to finite
density by assuming a rational form for a function defined in Laplace space,
the coefficients being determined by simple physical requirements. Fourier
transform in terms of reverse Bessel polynomials constitute the mathematical
framework of this approximation, from which an analytical expression for the
static structure factor is obtained. In its most elementary form, the method
recovers the solution of the Percus-Yevick closure to the Ornstein-Zernike
equation for hyperspheres at odd dimension. The present formalism allows one to
go beyond by yielding solutions with thermodynamic consistency between the
virial and compressibility routes to any desired equation of state. Excellent
agreement with available computer simulation data at $d=5$ and $d=7$ is
obtained. As a byproduct of this study, an exact and explicit polynomial
expression for the intersection volume of two identical hyperspheres in
arbitrary odd dimensions is given.
