Martínez-Finkelshtein, Andrei; Moreno-Balcázar, Juan J.; Pijeira-Cabrera, Héctor Strong asymptotics for Gegenbauer-Sobolev orthogonal polynomials. (English) Zbl 0895.33003 J. Comput. Appl. Math. 81, No. 2, 211-216 (1997). Consider the Gegenbauer-Sobolev inner product \[ (f,g)_S=\int_{-1}^1f(x)g(x)(1-x^2)^{\alpha-\frac{1}{2}}dx+ \lambda\int_{-1}^1f'(x)g'(x)(1-x^2)^{\alpha-\frac{1}{2}}dx, \] where \(\alpha>-\frac{1}{2}\) and \(\lambda\geq 0\). For \(\lambda=0\) this leads to the well-known classical Gegenbauer polynomials. In the case \(\lambda >0\) the corresponding orthogonal polynomials are called Gegenbauer-Sobolev polynomials. These polynomials and their algebraic properties have been studied by T.E. Pérez in her Ph.D. thesis “Polinomios orthogonales respecto a productos de Sobolev : el caso continuo”. In this paper the authors study the asymptotic behaviour of the monic Gegenbauer-Sobolev polynomials. The asymptotics of the zeros and norms of these orthogonal polynomials are also considered. The zeros of these Gegenbauer-Sobolev polynomials were studied earlier by H. G. Meijer [Coherent pairs and zeros of Sobolev-type orthogonal polynomials, Indag. Math., New Ser. 4, 163-176 (1993; Zbl 0784.33004)] and also by T. E. Pérez in her thesis [see also F. Marcellán, T. E. Pérez and M. A. Piñar, Gegenbauer-Sobolev orthogonal polynomials, in : A. Cuyt (ed.), Proc. Conf. on Nonlinear Numerical Methods and Rational Approximation II, Kluwer Academic Publishers, Dordrecht, 71-82 (1994; Zbl 0815.33008)]. T. E. Pérez has already shown that the zeros of the Gegenbauer-Sobolev polynomials are real and simple, and that they interlace with the zeros of the classical Gegenbauer polynomials. Furthermore, when \(\alpha\geq\frac{1}{2}\) all zeros are contained in \([-1,1]\) and for \(-\frac{1}{2}<\alpha<\frac{1}{2}\) there is at most one pair of zeros symmetric with respect to the origin outside the interval \([-1,1]\). In this paper it is also shown that the zeros are dense in \([-1,1]\). Reviewer: Roelof Koekoek (Delft) Cited in 9 Documents 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 Keywords:Gegenbauer-Sobolev orthogonal polynomials; asymptotics Citations:Zbl 0784.33004; Zbl 0815.33008 PDFBibTeX XMLCite \textit{A. Martínez-Finkelshtein} et al., J. Comput. Appl. Math. 81, No. 2, 211--216 (1997; Zbl 0895.33003) Full Text: DOI References: [1] Alfaro, M.; Marcellán, F.; Rezola, M. L., Estimates for Jacobi-Sobolev type orthogonal polynomials (1996), Universidad de Zaragoza, Preprint · Zbl 0888.33006 [2] Iserles, A.; Koch, P. E.; Nørsett, S.; Sanz-Serna, J. M., On polynomials orthogonal with respect to certain Sobolev inner products, J. Approx. Theory, 65, 151-175 (1991) · Zbl 0734.42016 [3] López, G.; Marcellán, F.; Van Assche, W., Relative asymptotics for polynomials orthogonal with respect to a discrete Sobolev inner product, Constr. Approx., 11, 107-137 (1995) · Zbl 0840.42017 [4] Marcellán, F.; Pérez, T. E.; Piñar, M. A., Gegenbauer-Sobolev orthogonal polynomials, (Cuyt, A., Proc. Conf. on Nonlinear Numerical Methods and Rational Approximation II (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 71-82 · Zbl 0815.33008 [5] Marcellán, F.; Ronveaux, A., Orthogonal polynomials and Sobolev inner products: a bibliography (1995), Facultés Universitaires N.D. de la Paix: Facultés Universitaires N.D. de la Paix Namur, Preprint [6] Marcellán, F.; Van Assche, W., Relative asymptotics for orthogonal polynomials, J. Approx. Theory, 72, 193-209 (1993) · Zbl 0771.42014 [7] Meijer, H. G., Coherent pairs and zeros of Sobolev-type orthogonal polynomials, Indag. Math. N.S., 4, 2, 163-176 (1993) · Zbl 0784.33004 [8] Pérez, T. E., Polinomios ortogonales respecto a productos de Sobolev: el caso continuo, (Ph.D. Thesis (1994), Departamento de Matemática Aplicada, Universidad de Granada) [9] Szegö, G., Orthogonal Polynomials, (Amer. Math. Soc. Colloq. Publ., 23 (1975), Amer. Math. Soc: Amer. Math. Soc Providence, RI) · JFM 65.0278.03 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.