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The remarkable properties of the associated Romanovski functions

For and with n − m − q ⩾ 0 a non-negative integer, we introduce a differential equation for the associated Romanovski functions R(q, β)n, m(x), so that its form remains invariant under the transformation n → 2q − n − 1. Introducing the Rodrigues formula for both regions and , we obtain simultaneously and separately realization of the laddering equations with respect to the labels n and m by means of two pairs of ladder operators. Also, the square integrability and orthogonality of the functions are considered in cases β ≠ 0 and β = 0 with respect to the weight functions and (1 + x2)−1 − q, respectively. Exponential generating functions for the infinite sequences with the same n in both regions and as well as for the infinite ones with given even and odd values of n − m − q are separately calculated. As an application to supersymmetric quantum mechanics, we show that there exist four different methods for generating the spectrum of the noncentral hyperbolic Scarf potential.