Connections between Romanovski and other polynomials

Hans Weber 1
  • 1 University of Virginia


A connection between Romanovski polynomials and those polynomials that solve the one-dimensional Schrödinger equation with the trigonometric Rosen-Morse and hyperbolic Scarf potential is established. The map is constructed by reworking the Rodrigues formula in an elementary and natural way. The generating function is summed in closed form from which recursion relations and addition theorems follow. Relations to some classical polynomials are also given.

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  • [1] E.J. Routh: “On some properties of certain solutions of a differential equation of second order”, Proc. London Math. Soc., Vol. 16, (1884), pp. 245–261.

  • [2] V. Romanovski: “Sur quelques classes nouvelles de polynomes orthogonaux”, C. R. Acad. Sci. Paris, Vol. 188, (1929), pp. 1023–1025.

  • [3] N. Cotfas: “Systems of orthogonal polynomials defined by hypergeometric type equations with application to quantum mechanics”, Cent. Eur. J. Phys., Vol. 2, (2004), pp. 456–466; “Shape invariant hypergeometric type operators with application to quantum mechanics”, Preprint: arXiv:math-ph/0603032.

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  • [5] A. Raposi, H.J. Weber, D. Alvarez-Castillo and M. Kirchbach: “Romanovski polynomials in selected physics problems”, Cent. Eur. J. Phys., to be published.

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  • [7] G.B. Arfken and H.J. Weber: Mathematical Methods for Physicists, 6th ed., Elsevier-Academic Press, Amsterdam, 2005.


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