We consider the application of the line strength formula recently derived by Watson (2006 J. Phys. B: At. Mol. Opt. Phys. 39 L291) to transitions between states of high principal quantum number in hydrogenic atoms and ions (Rydberg–Rydberg transitions). Apparent difficulties in the implementation of this formula are overcome by the use of recurrence relations derived by the ladder operator technique of Infeld and Hull (1951 Rev. Mod. Phys. 23 21), and set out in an earlier paper by the present author (2006 J. Phys. B: At. Mol. Opt. Phys. 39 2641). The use of the McLean–Watson formula for such cases is illustrated by the determination of the radiative lifetimes for levels with n ≈ 1000 and comparison of present results with approximate formulae. Interest in this work on the radial matrix elements for large n and n' is related both to measurements of radio recombination lines from tenuous space plasmas, e.g. Stepkin et al (2007 Mon. Not. R. Astron. Soc. 374 852) and to the calculation of Stark broadening for such spectra, e.g. Gigosos et al (2007 Astron. Astrophys. 466 1189), Stambulchik et al (2007 Phys. Rev. E 75 016401) and Stambulchik and Maron (2008 J. Phys. B: At. Mol. Opt. Phys. 41 095703). In addition, we discuss the question of inaccuracy caused by the omission of fine structure in such calculations, and the numerical stability of the recurrence relations used to implement the line strength formulae.