Recurrence formulae for arbitrary hydrogenic radial matrix elements are obtained in the Dirac form of relativistic quantum mechanics. Our approach is inspired by the relativistic extension of the second hypervirial method that has been succesfully employed to deduce an analogous relationship in non-relativistic quantum mechanics. We first obtain the relativistic extension of the second hypervirial and then the relativistic recurrence relation. Furthermore, we use this relation to deduce relativistic versions of the Pasternack-Sternheimer rule and of the virial theorem.