An algorithm for the recursive evaluation of expectation values for a given state of the relativistic one-electron atom with a point-like nucleus of charge number Z is proposed. In contrast to previous approaches, the present algorithm is based on only a single recurrence relation, which is a recurrence relation for a special type of generalized hypergeometric series. Two sequences of values of such series are generated. Finally, the members of these two sequences are assembled to yield the expectation values. The algorithm can be considered as a relativistic analogue of the Kramers - Pasternack recursion for expectation values for states of the non-relativistic hydrogen-like atom. As an application closed-form expressions for , were derived. In addition, numerical values for are given for some representative states of one-electron atoms with Z = 1, 80 and 137.