Series solution of the differential equation determining the nth-shell one-electron density of a bare Coulomb problem in quantum physics

The differential equation determining the nth-shell one-electron density of a bare Coulomb problem is analytically investigated by using the differential function method and extended homogeneous balance idea. New series solution of the differential equation has been obtained in quantum physics for the first time. The explicit series analytical expression for the mathematical characteristic of the bare Coulomb problem in the density-functional theory can further describe the dynamical behaviors of the nth-shell one-electron density.