Algebraic relations from finite-nuclear-mass effects to test atomic transition rates

General algebraic relations are derived which provide a stringent test of the accuracy of n-photon electric dipole transition rates when mass-polarization effects are included. They are a generalization of the well-known equivalence of the length, velocity, and acceleration forms of the transition matrix element that follows from gauge invariance. The algebraic relations connect the coefficients in a power series in powers of mu/M for the three gauges, where M is the nuclear mass and mu is the electron reduced mass. These relations also provide a stringent test of the leading infinite-mass term, a quantity that must be calculated with sufficient accuracy for the higher-order terms in powers of mu/M to be correct. As a check, the length-velocity algebraic relations are used to test the accuracy of high-precision calculations for both one-photon (1s2p 1P -1s2 1S and 1s2p 3P -1s2s 3S) and two-photon (1s2s 1S -1s2 1S) decay for the heliumlike ions with Z = 2-10.