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In mathematics, Romanovski polynomials are a family of real polynomials related to Jacobi polynomials solving a special hypergeometric differential equation, detailed below. They depend on two parameters and have first been considered by Sir Edward John Routh〔E. J. Routh, "On some properties of certain solutions of a differential equation of second order", Proc. London Math. Soc. 16 (1884) 245.〕 in 1884, to be rediscovered later within the context of probability distribution functions in statistics by Vsevolod Romanovski 〔V. Romanovski, "Sur quelques classes nouvelles de polynomes orthogonaux", C. R. Acad. Sci. Paris, 188 (1929) 1023.〕 (also Romanovsky) in 1929. They are referred to in the mathematics literature as "Romanovski",〔A. P. Raposo, H. J. Weber, D. E. Alvarez Castillo, and M. Kirchbach, "Romanovski polynomials in selected physics problems", Centr. Eur. Jour. Phys. 5 (3) (2007) 253.〕 or "Routh−Romanovski" 〔M. Masjed-Jamei, F. Marcellan, and E. J. Huertas, "A finite class of orthogonal functions by Routh–Romanovsky polynomials, Complex Variables and Elliptic Equations: An Int. Jour. bf 59}(2) (2014) 162.〕 polynomials. In some contrast to the standard classical orthogonal polynomials, the polynomials under consideration differ, in so far as for arbitrary parameters only ''a finite number of them are orthogonal'', as discussed in more detail below. ==The differential equation for the Romanovski polynomials== The Romanovski polynomials solve the following version of the hypergeometric differential equation Curiously, they have been omitted from the standard text books on special functions in mathematical physics 〔.〕〔A. F. Nikiforov and V. B. Uvarov, "Special Functions of Mathematical Physics", Birkhaeuser Verlag, Basel, 1988.〕 and in mathematics〔G. Szego, "Orthogonal Polynomials", American Math. Soc., Vol. XXIII, Prov., RI, 1939.〕〔M. E. H. Ismail, "Classical and Quantum Orthogonal Polynomials in One Variable", Cambridge Univ. Press, 2005.〕 and have only a relatively scarce presence elsewhere in the mathematical literature.〔P. A. Lesky, "Endliche und unendliche Systeme von kontinuierlichen klassischen Orthogonalpolynomen", Z. Angew. Math. Mech. 76 (1996) 181.〕〔R. Askey, "Beta integrals and the associated orthogonal polynomials", in Number Theory, Madras, 1987, vol. 1395 of Lecture Notes in Math. Springer, Berlin, p. 84.〕〔A. Zarzo-Altarejos, "Differential Equations of the Hypergeometric Type", (in Spanish), Ph. D. thesis, Faculty of Science, University of Granada (1995).〕 The weight functions are they solve Pearson's differential equation that assures the self-adjointness of the differential operator of the hypergeometric ordinary differential equation. For =0 and negative values, the weight function of the Romanovski polynomials takes the shape of the Cauchy distribution, whence the associated polynomials are also denoted as Cauchy polynomials〔N. S. Witte and P. J. Forrester, "Gap probabilities in finite and scaled Cauchy random matrix ensembles", Nonlinearity 13 (2000) 1965.〕 in their applications in random matrix theory.〔P. J. Forrester, "Log-Gases and Random Matrices", London Math. Soc. Monographs, Princeton University Press, 2010.〕 The Rodrigues formula specifies the polynomial as (x)s(x)^n \right ), \quad 0\leq n, |}} where is a normalization constant. This constant is related to the coefficient of the term of degree in the polynomial by the expression ,\quad \lambda_n=-n\left(t^_n(n-1)s^(x)\right), |}} which holds for ≥ 1. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Romanovski polynomials」の詳細全文を読む スポンサード リンク
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