Understanding the analysis of wave function

We obtained the order $\rho$ and the type $\sigma$ of wave function with power-law potentials, and found that the order and the type are compatible with the condition $|\sigma|\rho =\sqrt{v_m}$ except to $m=-2$. At the same time, we ansatz that the wave function satisfaction relation $\psi(r)=f(r)\exp[g(r)]$, where $f(r),g(r)$ are polynomials, and the power of $g(r)$ is no more than the order $\rho$.