[en] On the basis of the original definition and analysis of the vector operator by Pauli (1926 Z. Phys. 36 336–63), and further developments by Flamand (1966 J. Math. Phys. 7 1924–31), and by Becker and Bleuler (1976 Z. Naturforsch. 31a 517–23), we consider the action of the operator on both spherical polar and parabolic basis state wave functions, both with and without direct use of Pauli’s identity (Valent 2003 Am. J. Phys. 71 171–75). Comparison of the results, with the aid of two earlier papers (Hey 2006 J. Phys. B: At. Mol. Opt. Phys. 39 2641–64, Hey 2007 J. Phys. B: At. Mol. Opt. Phys. 40 4077–96), yields a convenient ladder technique in the form of a recurrence relation for calculating the transformation coefficients between the two sets of basis states, without explicit use of generalized hypergeometric functions. This result is therefore very useful for application to Stark effect and impact broadening calculations applied to high-n radio recombination lines from tenuous space plasmas. We also demonstrate the versatility of the Runge–Lenz–Pauli vector operator as a means of obtaining recurrence relations between expectation values of successive powers of quantum mechanical operators, by using it to provide, as an example, a derivation of the Kramers–Pasternack relation. It is suggested that this operator, whose potential use in Stark- and Zeeman-effect calculations for magnetically confined fusion edge plasmas (Rosato, Marandet and Stamm 2014 J. Phys. B: At. Mol. Opt. Phys. 47 105702) and tenuous space plasmas ( H II regions) has not been fully explored and exploited, may yet be found to yield a number of valuable results for applications to plasma diagnostic techniques based upon rate calculations of atomic processes. (paper)