Radial distribution function for hard spheres

The radial distribution function g(r) provided by the solution of the Percus-Yevick (PY) equation for hard spheres is rederived in terms of the simplest Pad\'e approximant of a function defined in the Laplace space that is consistent with the following physical requirements: g(r) is continuous for r>1, the isothermal compressibility is finite, and the zeroth- and first-order coefficients in the density expansion of g(r) must be exact. An explicit expression for the solution of the generalized mean-spherical approximation (GMSA) is obtained as a simple extension involving two new parameters, which are determined by imposing two conditions: (i) the virial and the compressibility routes to the equation of state agree consistently, and (ii) this equation of state coincides with that of Carnahan and Starling [J. Chem. Phys. 51, 635 (1969)]. The second- and third-order coefficients in the density expansion of g(r) given by the GMSA are compared with the exact ones and with those given by the PY equation.