In recent work, Howard and March [J. Phys. A: Math. Gen. 37 (2003) 965] have proposed a relativistic density functional theory of harmonically confined independent fermions in terms of a difference equation. Here, a corresponding relativistic difference equation is proposed and solved for hydrogenlike atomic ions for arbitrary atomic number. For a point nucleus, the singularity predicted by Dirac's one-electron wave equation is thereby avoided.