This constant is related to the coefficient cn of the term of degree n in the polynomial R(α,β)n(x) by the expression which holds for n ≥ 1.
Since the cited authors discuss the non-hermitian (complex) orthogonality conditions only for real Jacobi indexes the overlap between their analysis and definition (8) of Romanovski polynomials exists only if α = 0.
Note that we have chosen the normalization constants Nn = 1, which is equivalent to making a choice of the coefficient of highest degree in the polynomial, as given by equation (5).
This is the case of a version of equation (1) that has been independently encountered anew within the context of the exact solubility of the quantum mechanical problem of the trigonometric Rosen–Morse potential and reported in Compean & Kirchbach (2006).
[17] There, the polynomial parameters α and β are no longer arbitrary but are expressed in terms of the potential parameters, a and b, and the degree n of the polynomial according to the relations, Correspondingly, λn emerges as λn = −n(2a + n − 1), while the weight function takes the shape Finally, the one-dimensional variable, x, in Compean & Kirchbach (2006)[17] has been taken as where r is the radial distance, while
In Compean & Kirchbach[17] it has been shown that the family of Romanovski polynomials corresponding to the infinite sequence of parameter pairs, is orthogonal.