Expectation Values in Relativistic Coulomb Problems

We evaluate the matrix elements 〈Orp〉, where O = {1, β, iαnβ} are the standard Dirac matrix operators and the angular brackets denote the quantum-mechanical average for the relativistic Coulomb problem, in terms of generalized hypergeometric functions 3F2 (1) for all suitable powers. Their connections with the Chebyshev and Hahn polynomials of a discrete variable are emphasized. As a result, we derive two sets of Pasternack-type matrix identities for these integrals, when p → −p − 1 and p → −p − 3, respectively. Some applications to the theory of hydrogenlike relativistic systems are reviewed.