An analytical expression for the first coordination cell of the radial distribution function (RDF) of the hard-sphere fluid is derived. It is based on a series expansion of the analytical expression of the Percus-Yevick solution of the RDF of the hard-sphere fluid derived by Chang and Sandler [J. Chang, S.I. Sandler, Mel. Phys. 81 (1994) 745.]. The expansion is carried out both in terms of the packing fraction and in terms of the radial distance. Expressions are also obtained for the coordination number and its first and second derivatives as functions of radial distance and packing fraction. These expressions, which are useful in perturbation theory, are simpler to use than those obtained from the starting equation, while giving nearly the same results. They also provide close agreement with simulation data.