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Connections between Romanovski and other polynomials. (English) Zbl 1155.33008

The author starts from the orthogonal polynomials of hypergeometric type with \(\sigma (x)=1+x^2\) and weight \(w_l(x)=\sigma (x)^{-(l+a+1)}e^{-\alpha \, \text{cot}^{-1}x}\). By writting the Rodrigues formula \(P_l^{(a,\alpha )}(x)=\frac{1}{w_l(x)}\frac{d^l}{dx^l}[w_l(x)\, \sigma (x)^l]\) in the form \[ P_l^{(a,\alpha )}(x)=\frac{1}{w_l(x)}\frac{d^{l-\nu }}{dx^{l-\nu }} [\sigma (x)^{l-\nu }w_l(x)Q_\nu ^{(\alpha ,-a)}(x)] \] with \(\nu =0,1,\dots,l\), the author defines inductively the complementary polynomials \(Q_\nu ^{(\alpha ,a)}(x)\). Similar to the classical orthogonal polynomials, the \(Q_\nu ^{(\alpha ,a)}(x)\) appear as solutions of a Sturm-Liouville ordinary second-order differential equation and obey Rodrigues formulas themselves. These real orthogonal polynomials and their nontrivial orthogonality properties are closely related to Romanovski polynomials and those polynomials that solve the one-dimensional Schrödinger equation with the trigonometric Rosen-Morse and hyperbolic Scarf potential. Relations to some classical polynomials are also given.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C47 Other special orthogonal polynomials and functions
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References:

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[2] V. Romanovski: “Sur quelques classes nouvelles de polynomes orthogonaux”, C. R. Acad. Sci. Paris, Vol. 188, (1929), pp. 1023-1025.; · JFM 55.0915.03
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