Mean values of powers of the radius for hydrogenic electron orbits

The general formulas available in the literature for calculating the time average of powers of the orbital radius of the electron in hydrogenic atoms or ions, 〈r^s〉, are of a degree about twice that of the resulting explicit expressions. In this paper, general formulas are given which are of the right degree and which are written in a suitable form for direct calculation of explicit formulas for specific values of s. Explicit expressions for 〈r^s〉 are given for -8≤s≤5 It is shown that formulas for positive and negative powers are connected by the formula 〈r^-(p+2)〉=(2Z / na₀)^2p+1[(2l-p)! / (2l+p+1)!]〈r^p-1〉, where p is any integer or zero.