On asymptotic properties of Freud–Sobolev orthogonal polynomials

16 pages, no figures.-- MSC2000 codes: 33C45; 33C47; 42C05.MR#: MR2016838 (2005e:33004)Zbl#: Zbl 1043.33005In this paper we consider a Sobolev inner product $(f,g)_S=\int fg\,d\mu+ \lambda \int f'g'\,d\mu (*)$, and we characterize the measures μ for which there exists an algebraic relation between the polynomials, {Pn}, orthogonal with respect to the measure μ and the polynomials, {Qn}, orthogonal with respect to (*), such that the number of involved terms does not depend on the degree of the polynomials. Thus, we reach in a natural way the measures associated with a Freud weight. In particular, we study the case $d\mu=e^{-x^4}dx$ supported on the full real axis and we analyze the connection between the so-called Nevai polynomials (associated with the Freud weight $e^{-x^4}$)and the Sobolev orthogonal polynomials Qn. Finally, we obtain some asymptotics for {Qn}.Research by first author (A.C.) partially supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under Grant BFM 2000 0015. Research by second author (F.M.) partially supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under Grant BFM2003 06335 C03 02, by INTAS Project 2000 272 and by the NATO collaborative Grant PST.CLG. 979738. Research by third author (J.J.M.-B.) partially supported by Junta de Andalucía, Grupo de Investigación FQM 0229, Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under Grant BFM 2001 3878 C02 02 and INTAS Project 2000 272.Publicad

Large Parameter Cases of the Gauss Hypergeometric Function

We consider the asymptotic behaviour of the Gauss hypergeometric function when several of the parameters a, b, c are large. We indicate which cases are of interest for orthogonal polynomials (Jacobi, but also Krawtchouk, Meixner, etc.), which results are already available and which cases need more attention. We also consider a few examples of 3F2-functions of unit argument, to explain which difficulties arise in these cases, when standard integrals or differential equations are not available.Comment: 21 pages, 4 figure

Asymptotics of Sobolev Orthogonal Polynomials for Coherent Pairs of Measures

AbstractStrong asymptotics for the sequence of monic polynomialsQn(z), orthogonal with respect to the inner product(f, g)S=∫f(x)g(x)dμ1(x)+λ∫ f′(x)g′(x)dμ2(x),λ>0,withzoutside of the support of the measureμ2, is established under the additional assumption thatμ1andμ2form a so-called coherent pair with compact support. Moreover, the asymptotic behaviour of the (square of) the norm (Qn, Qn)Sand of the zeros ofQnis obtained