An alternative approach to calculating the mean values rs¯ for relativistic hydrogen-like atoms

A direct approach for deriving the formulas of the mean values rs¯ of hydrogen-like atoms for an integer s is developed in relativistic quantum mechanics. Explicit formulas of rs¯ for |s|⩽6 are presented and their non-relativistic limits are investigated resulting in explicit formulas of rs¯ for hydrogen-like atoms in non-relativistic quantum mechanics. The forward and backward two-step uncoupled recursion relations of rs¯ are also constructed out. Formulas of rs¯ for |s|⩽6 are demonstrated satisfying both two-step uncoupled recursion relations and one-step coupled recursion relations.