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

The paper links Stevenson’s formally complex hypergeometric polynomials to the real-by-definition Romanovski-Routh (R-Routh) polynomials. The spectral problem for Stevenson’s second-order normal ordinary differential equation (NODE) was formulated in such a way that it could be re-used for the two SLPs associated with the ‘trigonometric Rosen-Morse’ (t-RM) potential on the finite interval and the implicit Milson potential on the line (both solvable by the R-Routh polynomials). Namely, the sought-for eigenfunction was required to represent the principal Frobenius solutions at both minus- and plus-infinity. We refer to these boundary conditions as the ‘dual-PFS’ problem. The exact solvability of the former SLP with the trigonometric Liouville potential was then proven by taking into account that the Romanovski-Routh polynomial of degree n must have exactly n real zeros as well as that the discrete energy spectrum in question had no upper bound. As the direct consequence of this proof, we then found that the mentioned d-PFS problem Stevenson’s NODE and therefore the second SLP associated with the Milson potential on the line were exactly solvable via the quasi-rational solutions (q-RSs) composed of the R-Routh polynomials with degree-dependent indexes.