New recursion relations of matrix elements of $r^λ$ and $βr^λ$ 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 $λ$ 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 --$r^λ$ or $βr^λ$, $β$ is a Dirac matrix-- in terms of the two previous consecutive elements. The results are useful in relativistic atomic calculations.

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