Exact irregular solutions to radial Schrödinger equation for the case of hydrogen-like atoms

This study propounds a novel methodology for obtaining the explicit/closed representation of the two linearly independent solutions of a large class of second order ordinary linear differential equation with special polynomial coefficients. The proposed approach is applied for obtaining the closed forms of regular and irregular solutions of the Coulombic Schrödinger equation for an electron experiencing the Coulomb force, and examples are displayed. The methodology is totally distinguished from getting these solutions either by means of associated Laguerre polynomials or confluent hypergeometric functions. Analytically, both the regular and irregular solutions spread in their radial distributions as the system energy increases from strongly negative values to values closer to zero. The threshold and asymptotic behavior indicate that the regular solutions have an $r^\ell$ dependence near the origin, while the irregular solutions diverge as $r^{-\ell-1}$. Also, the regular solutions drop exponentially in proportion to $r^{n-1}\exp(-r/n)$, in natural units, while the irregular solutions grow as $r^{-n-1}\exp(r/n)$. Knowing the closed form irregular solutions leads to study the analytic continuation of the complex energies, complex angular momentum, and solutions needed for studying bound state poles and Regge trajectories.