Overlapping of Lévai’s and Milson’s e-Tangent-Polynomial Potentials along Symmetric Curves

The paper examines common elements between Lévai’s and Milson’s potentials obtained by Liouville transformations of two rational canonical Sturm–Liouville equations (RCSLEs) with even density functions which are exactly solvable via Jacobi polynomials in a real or accordingly imaginary argument. We refer to the polynomial numerators given function as ‘tangent polynomial’ (TP) thereby term aforementioned ‘e-TP’. Special attention is overlap along symmetric curves represent different forms Ginocchio potential quantized Gegenbauer Masjed-Jamei polynomials, respectively. Our analysis reveals that actual interconnection parameters for these realizations much more complicated than one could expect based on striking resemblance quartic derived Lévai ‘averaged’ indexes.