Termination of the Coulomb radial series ($E_n=-\frac{m{K}^2}{2{\hslash }^2{n}^2}$)

For a negative Coulomb energy, remove the exponential decay and regular radial power to obtain a confluent-hypergeometric series. Its recurrence terminates only when $n=q+\ell +1$, with $q$ a nonnegative integer. Regularity at the origin and normalizability at infinity then give the hydrogenic energy levels. Without those physical boundary conditions the radial differential equation has other solutions.