On using vector spherical harmonics

In many astronomical works, the structure of vector fields is analyzed, such as the differences in the celestial object coordinates in catalogs or the celestial object velocities, by decomposition into vector spherical harmonics (VSH). This method has shown high efficiency in many studies, but, at the same time, comparing the results obtained by different authors can cause difficulties associated with different approaches to building the VSH system and even with their different designations. To facilitate this task, this paper provides a comparison of the three VSH systems most often used in works on astrometry and stellar astronomy.