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The complementary polynomials and the Rodrigues operator of classical orthogonal polynomials. (English) Zbl 1281.33006
The authors focus on the complementary polynomials defined by H. J. Weber in [Cent. Eur. J. Math. 5, No. 2, 415–427 (2007; Zbl 1124.33011)], here rewritten by means of the so-called Rodrigues operator introduced in a previous paper by R. S. Costas-Santos and F. Marcellán [Acta Appl. Math. 111, No. 1, 107–128 (2010; Zbl 1204.33011)], since a more general context is required by considering differential or difference or \(q\)-difference operators.
With respect to these complementary polynomials, this paper presents Rodrigues functional formulas, a Sturm-Liouville type equation and several informations about the corresponding generating function. In particular, this work extends the results obtained by Weber [loc. cit.] for the standard derivative operator.
MSC:
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
34B24 Sturm-Liouville theory
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