arXiv:0908.3021v5 [math-ph] 30 Sep 2009 RELATIVISTIC KRAMERS–PASTERNACK RECURRENCE RELATIONS SERGEI K. SUSLOV Abstract. Recently we have evaluated the matrix elements 〈Or p 〉, 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 3 F 2 (1) for all suitable powers and established two sets of Pasternack-type matrix identities for these integrals. The corresponding Kramers–Pasternack type three-term vector recurrence relations are derived here. 1. Introduction Recent experimental and theoretical progress has renewed interest in quantum electrodynamics of atomic hydrogenlike systems (see, for example, [22], [23], [27], [28], [36], [53], and [55] and references therein). In the last decade, the two-time Green’s function method of deriving formal expressions for the energy shift of a bound-state level of high-Z few-electron systems was developed [53] and numerical calculations of QED effects in heavy ions were performed with excellent agreement to current experimental data [22], [23], [55]. These advances motivate, among other technical things, investigation of the expectation values of the Dirac matrix operators between the bound-state relativistic Coulomb wave functions. Special cases appear in calculations of the magnetic dipole hyperfine splitting, the electric quadrupole hyperfine splitting, the anomalous Zeeman effect, and the relativistic recoil corrections in hydrogenlike ions (see, for example, [1], [52], [54], [57] and references therein). In the previous paper [57], the matrix elements 〈Or p 〉, where O = {1,β,iαnβ } are the standard Dirac matrix operators and the angular brackets denote the quantum-mechanical average for the rel- ativistic Coulomb problem, have been evaluated as sums of three special generalized hypergeometric functions 3 F 2 (1) (or Chebyshev polynomials of a discrete variable) for all suitable powers. As a result, two sets of Pasternack-type matrix symmetry relations for these integrals, when p →−p − 1 and p →−p − 3, have been derived. We concentrate on what are essentially radial integrals since, for problems involving spherical symmetry, one can reduce all expectation values to radial integrals by use of the properties of angular momentum. Our next goal is to establish relativistic analogs of the Kramers–Pasternack recurrence relation [31], [44], and [45] (some progress in this direction had been made in [2]). Here several three-term vector recurrence relations are obtained, which follow in a natural way from the well-known theory of classical orthogonal polynomials of a discrete variable [39] and basic matrix algebra. This paper is organized as follows. We review the nonrelativistic case in the next section and then present Date : August 26, 2018. 1991 Mathematics Subject Classification. Primary 81Q05. Secondary 33C20. Key words and phrases. The Dirac equation, hydrogenlike ions, expectation values, Hahn polynomials, Kramers– Pasternack recurrence relation. 1