The exact mathematical expression for an arbitrary nth-order stellar hydrodynamic equation is explicitly obtained depending on the central moments of the velocity distribution. In such a form the equations are physically meaningful, since they can be compared with the ordinary hy- drodynamic equations of compressible, viscous fluids. The equations are deduced without any particular assumptions about symmetries, steadi- ness, or particular kinematic behaviours, so that they can be used in their complete form, and for any order, in future works with improved observational data. Also, in order to work with a finite number of equa- tions and unknowns, which would provide a dynamic model for the stel- lar system, the nth-order equation is needed to investigate in a more general way the closure conditions, which may be expressed in terms of velocity distribution statistics, as it is shown in a case example.