Rediscovery of Routh Polynomials after Hundred Years in Obscurity

Abstract

The introductory part of the paper outlines misfortunate history of one of Routh’ most remarkable works. It points to an important distinction between infinitely many Routh polynomials forming a differential polynomial system (DPS) and their finite orthogonal subset referred to as ‘Romanovski-Routh’ (R-Routh) polynomials (Romanovski/pseudo-Jacobi polynomials in Lesky's classification scheme). It was shown that there are two infinite sequences of Routh polynomials without real roots (in contrast with R-Routh polynomials with all the roots located on the real axis). The factorization functions (FFs) formed by polynomials from these infinite sequences can be thus used to generate exactly solvable rational Darboux transforms of the canonical Sturm-Liouville equation (CSLE) quantized in terms of R-Routh polynomials with degree-dependent indexes. The current analysis is focused solely on the CSLE with the density function having two simple zeros on the opposite sides of the imaginary axis. A special attention is given to the limiting case of the density function with the simple poles at +i and -i when the given CSLE becomes translationally form-invariant and as a result the eigenfunctions of its rational Darboux transforms are expressible in term of a finite number of exceptional orthogonal polynomials (EOPs).