Two tool parameters, the beam propagation factor M² and kurtosis factor k, are used to analyse solutions of the paraxial scalar wave equation (for rectangular coordinates) i.e. the so-called the standard and elegant Hermite–Gaussian modes (SHG and EHG). Analytical expressions for M² and k are derived and the correspondence between them for each form of solution is discussed in detail. These two parameters reveal information about the solutions, which leads us to derive the asymptotic representation of EHG modes (i.e. in the near-field, for a large number of modes).