Ratio and Plancherel-Rotach asymptotics for Meixner-Sobolev orthogonal polynomials

13 pages, no figures.-- MSC codes: 42C05; 33C25; 39A10.MR#: MR1741786 (2000k:42032)Zbl#: Zbl 0984.42016We study the analytic properties of the monic Meixner-Sobolev polynomials $\{Q_n\}$ orthogonal with respect to the inner product involving differences $$(p,q)_S=\sum infty_{i=0}[p(i)q(i)+\lambda\Delta p(i)\Delta q(i)] {\mu (\gamma)_i\over i!},$$ $\gamma>0,\ 0<\mu<1$, where $\lambda\geq0,\ \Delta$ is the forward difference operator $(\Delta f(x)=f(x+1)-f(x))$ and $(\gamma)_n$ denotes the Pochhammer symbol. Relative asymptotics for Meixner-Sobolev polynomials with respect to Meixner polynomials is obtained. This relative asymptotics is also given for the scaled polynomials. Moreover, a zero distribution for the scaled Meixner-Sobolev polynomials and Plancherel-Rotach asymptotics for $\{Q_n\}$ are deduced.The work of E.G. has been partially supported by Dirección General de Enseñanza Superior (DGES) of Spain under Grant PB-96-0952. The work of F.M. is partially supported by PB96-0120-C03-01 and INTAS-93-0219 Ext. The work of J.J.M.-B. is partially supported by Junta de Andalucía, G.I. FQM0229.Publicad

Asymptotic properties of generalized Laguerre orthogonal polynomials

In the present paper we deal with the polynomials L(α,M,N) n (x) orthogonal with respect to the Sobolev inner product (p, q) = 1 Γ(α+1) Z ∞ 0 p(x)q(x) x α e −x dx + M p(0)q(0) + N p 0 (0)q 0 (0), N,M ≥ 0, α > −1, firstly introduced by Koekoek and Meijer in 1993 and extensively studied in the last years. We present some new asymptotic properties of these polynomials and also a limit relation between the zeros of these polynomials and the zeros of Bessel function Jα(x). The results are illustrated with numerical examples. Also, some general asymptotic formulas for generalizations of these polynomials are conjectured.Junta de AndalucíaDirección General de InvestigaciónUnión Europe

Inner products involving q-differences: the little q-Laguerre-Sobolev polynomials

22 pages, no figures.-- MSC codes: Primary 33C25; Secondary 33D45.-- Issue title: "Higher transcendental functions and their applications".MR#: MR1765938 (2001d:33018)Zbl#: Zbl 0957.33008In this paper, polynomials which are orthogonal with respect to the inner product \multline\langle p,r\rangle_S=\sum^\infty_{k=0}p(q^k)r(q^k) {(aq)^k(aq;q)_\infty\over(q;q)_k}\\ +\lambda\sum^\infty_{k=0} (D_qp)(q^k)(D_qr)(q^k){(aq)^k(aq;q)_\infty\over(q;q)_k},\endmultline where $D_q$ is the $q$-difference operator, $\lambda\geq0,\ 0<q<1$ and $0<aq<1$, are studied. For these polynomials, algebraic properties and $q$-difference equations are obtained as well as their relation with the monic little $q$-Laguerre polynomials. Some properties of the zeros of these polynomials are also deduced. Finally, the relative asymptotics $\{Q_n(x)/p_n(x;a )\}_n$ on compact subsets of {\bf C}\sbs[0,1] is given, where $Q_n(x)$ is the $n$th degree monic orthogonal polynomial with respect to the above inner product and $p_n(x;a )$ denotes the monic little $q$-Laguerre polynomial of degree $n$.E.G. wishes to acknowledge partial financial support by Dirección General de Enseñanza Superior (DGES) of Spain under Grant PB-96-0952. The research of F.M. was partially supported by DGES of Spain under Grant PB96-0120-C03-01 and INTAS Project 93-0219 Ext. J.J.M.B. also wishes to acknowledge partial financial support by Junta de Andalucía, Grupo de Investigación FQM 0229.Publicad

Asymptotics and zeros of Sobolev orthogonal polynomials on unbounded supports

In this paper we present a survey about analytic properties of polynomials orthogonal with respect to a weighted Sobolev inner product such that the vector of measures has an unbounded support. In particular, we are focused in the study of the asymptotic behaviour of such polynomials as well as in the distribution of their zeros. Some open problems as well as some new directions for a future research are formulated.Comment: Changed content; 34 pages, 41 reference

k-Coherence of measures with non-classical weights

7 pages, no figures.-- MSC2000 codes: Primary 42C05; Secondary 33C25.MR#: MR1882617 (2003b:42042)The concept of k-coherence of two positive measures μ1 and μ2 is useful in the study of the Sobolev orthogonal polynomials. If μ1 or μ2 are compactly supported on R then any 0-coherent pair or symmetrically 1-coherent pair (μ1, μ2) must contain a Jacobi measure (up to affine transformation). Here examples of k-coherent pairs (k ≥ 1) when neither μ1 nor μ2 are Jacobi are constructed.Research of F. Marcellán supported by Dirección General de Investigación(Ministerio de Ciencia y Tecnología) of Spain under grant BFM2000-0206-C04-01. Research of A. Martínez-Finkelshtein partially supported by INTAS project 2000-272, a research grant of Dirección General de Enseñanza Superior (DGES) of Spain, project code PB95-1205, and by Junta de Andalucía, Grupo de Investigación FQM 0229. Research of J. J. Moreno-Balcázar partially supported by Junta de Andalucía, Grupo de Investigación FQM 0229, and INTAS project 2000-272.Publicad

Concerning asymptotic behavior for extremal polynomials associated to nondiagonal sobolev norms

Let ℙ be the space of polynomials with complex coefficients endowed with a nondiagonal Sobolev norm || · ||W 1, p (Vμ), where the matrix V and the measure μ constitute a p -admissible pair for 1 ≤ p ≤ ∞. In this paper we establish the zero location and asymptotic behavior of extremal polynomials associated to || · ||W 1, p (Vμ), stating hypothesis on the matrix V rather than on the diagonal matrix appearing in its unitary factorizationAna Portilla and Eva Tourís are supported in part by a grant from Ministerio de Ciencia e Innovación (MTM 2009-12740-C03-01), Spain. Yamilet Quintana is supported in part by the Research Sabbatical Fellowship Program (2011-2012) from Universidad Simón Bolvar, Venezuela. Ana Portilla, José M. Rodríguez, and Eva Tourís are supported in part by two grants from Ministerio de Ciencia e Innovación (MTM2009-07800 and MTM2008-02829-E), Spain. José M. Rodríguez is supported in part by a grant from CONACYT (CONACYT-UAGI0110/62/10 FON.INST.8/10), Méxic

Recent trends in orthogonal polynomials and their applications

29 pages, 1 figure.-- MSC2000 codes: 42C05, 33C45.-- Contributed to: XVII CEDYA: Congress on differential equations and applications/VII CMA: Congress on applied mathematics (Salamanca, Spain, Sep 24-28, 2001).MR#: MR1873645 (2002i:42031)Zbl#: Zbl 1026.42025In this contribution we summarize some new directions in the theory of orthogonal polynomials. In particular, we emphasize three kinds of orthogonality conditions which have attracted the interest of researchers from the last decade to the present time. The connection with operator theory, potential theory and numerical analysis will be shown.This work has been supported by Dirección General de Investigación (MCyT) of Spain under grant BHA2000-0206-C04-01 and INTAS project 2000-272. J. Arvesú was partially supported by the Dirección General de Investigación (Comunidad Autónoma de Madrid).Publicad