Long Journey from Stevenson’s Formally Complex Hypergeometric Polynomials to Real-by-Definition Romanovski-Routh Polynomials

The paper reexamines Stevenson’s technique for solving Schrödinger’s “Kepler problem” in a spherical space in terms of formally complex hypergeometric polynomials. A certain advantage has been achieved by reformulating the genetic 'dual principal Fro-benius solution’ (d-PFS) problem as the Dirichlet problem for the given second-order or-dinary differential equation (ODE) rewritten in its 'prime' form. It was demonstrated that the cited polynomials match Askey’s hypergeometric expressions for the re-al-by-definition Romanovski/pseudo-Jacobi polynomials (‘Romanovski-Routh’ polyno-mials in our terms). The formulated Dirichlet problem was then reduced to the two more specific cases representing the Sturm-Liouville problems (SLPs) with infinite and respec-tively finite discrete energy spectra. The exact solvability of the former SLP (with the Li-ouville potential represented by the ‘trigonometric Rosen-Morse’ potential) was proven by taking into account that the Romanovski-Routh polynomial of degree n must have exactly n real zeros (with no upper bound for the eigenvalues). As the direct consequence of this proof, we then found that the mentioned d-PFS problem in general and therefore the second SLP with the finite discrete energy spectrum are exactly solvable via qua-si-rational solutions (q-RSs) composed of the Romanovski/Routh polynomials with de-gree-dependent indexes.