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The semiclassical Sobolev orthogonal polynomials: a general approach. (English) Zbl 1215.33005
A linear functional $${\mathbf u}$$ (defined on the space of polynomials $$\mathbb P)$$ is called a semiclassical functional if it satisfies a distributional equation with polynomial coefficients:
${\mathcal D}(\varphi{\mathbf u})=\psi{\mathbf u},$ where $${\mathcal D}$$ is the differential, or the difference or the $$q$$-difference operator. The polynomial sequence $$(Q_n^{(\lambda)})$$ is called a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product
$\langle p,r\rangle _S=\langle{\mathbf u},pr\rangle+\lambda\langle{\mathbf u},{\mathcal D} p{\mathcal D} r\rangle.$ The authors get algebraic and differential/difference properties for such polynomials as well as algebraic relations between them and the polynomial sequence orthogonal with respect to the semiclassical functional $${\mathbf u}$$. In particular, a linear operator $${\mathcal J}$$ such that
$\langle{\mathcal J}p, r\rangle_S=\langle p,{\mathcal J}r\rangle_S,\quad p,r\in\mathbb P,$ is constructed.
The main goal of the article is to give a general approach to the study of the polynomials orthogonal with respect to the above nonstandard inner product regardless of the type of operator $${\mathcal D}$$ considered. Finally, the results are illustrated by applying them to some known families of Sobolev orthogonal polynomials (Jacobi-Sobolev and $$\Delta$$-Meixner-Sobolev ones) as well as to some new ones introduced for the first time, such as $$q$$-Freud-Sobolev ones and another family related to a 1-singular semiclassical functional considered by Medem.

##### 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
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##### References:
  Area, I.; Godoy, E.; Marcellán, F., Classification of all $$\Delta$$-coherent pairs, Integral transforms spec. funct., 9, 1, 1-18, (2000) · Zbl 0972.42017  Area, I.; Godoy, E.; Marcellán, F., $$q$$-coherent pairs and $$q$$-orthogonal polynomials, Appl. math. comput., 128, 2-3, 191-216, (2002) · Zbl 1020.33005  Chihara, T.S., An introduction to orthogonal polynomials, (1978), Gordon and Breach New York · Zbl 0389.33008  Costas-Santos, R.S.; Marcellán, F., Second structure relation for $$q$$-semiclassical polynomials of the Hahn tableau, J. math. anal. appl., 329, 1, 206-228, (2007) · Zbl 1113.33022  Hahn, W.M., Über orthogonalpolynomen, die $$q$$-differentialgleichungen genügen, Math. Z., 39, 636-638, (1949)  Iserles, A.; Koch, P.E.; Nørsett, S.P.; Sanz-Serna, J.M., On polynomials orthogonal with respect to certain Sobolev inner products, J. approx. theory, 65, 151-175, (1991) · Zbl 0734.42016  Kheriji, L., An introduction to the $$H_q$$-semiclassical orthogonal polynomials, Methods appl. anal., 10, 387-412, (2003) · Zbl 1058.33018  R. Koekoek, R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its $$q$$-analogue, volume 98-17, Reports of the Faculty of Technical Mathematics and Informatics, Delft, The Netherlands, 1998.  Lewis, D.C., Polynomial least square approximations, Amer. J. math., 69, 273-278, (1947) · Zbl 0033.35603  Marcellán, F.; Alfaro, M.; Rezola, M.L., Orthogonal polynomials on Sobolev spaces: old and new directions, J. comput. appl. math., 48, 113-131, (1993) · Zbl 0790.42015  Marcellán, F.; Moreno-Balcázar, J.J., Asymptotics and zeros of Sobolev orthogonal polynomials on unbounded supports, Acta appl. math., 94, 163-192, (2006) · Zbl 1137.42312  Maroni, P., Une théorie algebrique des polynômes orthogonaux. applications aux polynômes orthogonaux semiclassiques, (), 98-130 · Zbl 0944.33500  Maroni, P., Variations around classical orthogonal polynomials. connected problems, J. comput. appl. math., 48, 133-155, (1993) · Zbl 0790.33006  Maroni, P., Semi-classical character and finite-type relations between polynomial sequences, Appl. numer. math., 31, 3, 295-330, (1999) · Zbl 0962.42017  Martínez-Finkelshtein, A., Asymptotic properties of Sobolev orthogonal polynomials, J. comput. appl. math., 99, 491-510, (1998) · Zbl 0933.42013  Martínez-Finkelshtein, A., Analytic aspects of Sobolev orthogonal polynomials revisited, J. comput. appl. math., 127, 255-266, (2001) · Zbl 0971.33004  Medem, J.C., A family of singular semi-classical functionals, Indag. math., 13, 3, 351-362, (2002) · Zbl 1031.42025  Meijer, H.G., A short history of orthogonal polynomials in a Sobolev space I. the non-discrete case, Nieuw arch. wiskd., 14, 93-112, (1996) · Zbl 0862.33001  Meijer, H.G., Determination of all coherent pairs, J. approx. theory, 89, 321-343, (1997) · Zbl 0880.42012  T.E. Pérez, Polinomios ortogonales respecto a productos de Sobolev: el caso continuo, Doctoral Dissertation, Universidad de Granada, 1994.  Sfaxi, R.; Marcellán, F., Second structure relation for semiclassical orthogonal polynomials, J. comput. appl. math., 200, 2, 537-554, (2007) · Zbl 1125.33008
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