New recursion relations of matrix elements of $r^\lambda$ and $\beta r^\lambda$ between relativistic hydrogenic eigenstates

We determine exact recurrence relations which help in the evaluation of matrix elements of powers of the radial coordinate between Dirac relativistic hydrogenic eigenstates. The power $\lambda$ can be any complex number as long as the corresponding term vanishes faster than $r^{-1}$ as $r \to \infty$. These formulas allow determining recursively any matrix element of radial powers --$r^\lambda$ or $\beta r^\lambda$, $\beta$ is a Dirac matrix-- in terms of the two previous consecutive elements. The results are useful in relativistic atomic calculations.