What is … \ldots\ a multiple orthogonal polynomial?

This is an extended version of our note in the Notices of the American Mathematical Society 63 (2016), no. 9, in which we explain what multiple orthogonal polynomials are and where they appear in various applications.Comment: 5 pages, 2 figure

Strong asymptotics for Sobolev orthogonal polynomials

14 pages, no figures.-- MSC1991 codes: 42C05, 33C25.Zbl 0937.42011In this paper we obtain the strong asymptotics for the sequence of orthogonal polynomials with respect to the inner product \langle f,g,\rangle=\sum_{k=0}\sp n \int_{\Delta_k} f\sp (k)}(x)g\sp (k)}(x) d\mu_k(x), where \{\mu_k\}_k=0\sp m, with m ∈ Z+ are measures supported on [−1,1] which satisfy Szegö's condition.Research by first author (A.M.F.) was partially supported by a research grant from Dirección General de Enseñanza Superior (DGES) of Spain, project code PB95-1205, a research grant from the European Economic Community, INTAS-93-219-ext, and by Junta de Andalucía, Grupo de Investigación FQM 0229.Publicad

Shannon entropy of symmetric Pollaczek polynomials

We discuss the asymptotic behavior (as $n\to \infty$) of the entropic integrals $$E_n= - \int_{-1}^1 \log \big(p^2_n(x) \big) p^2_n(x) w(x) d x,$$ and $$F_n = -\int_{-1}^1 \log (p_n^2(x)w(x)) p_n^2(x) w(x) dx,$$ when $w$ is the symmetric Pollaczek weight on $[-1,1]$ with main parameter $\lambda\geq 1$, and $p_n$ is the corresponding orthonormal polynomial of degree $n$. It is well known that $w$ does not belong to the Szeg\H{o} class, which implies in particular that $E_n\to -\infty$. For this sequence we find the first two terms of the asymptotic expansion. Furthermore, we show that $F_n \to \log (\pi)-1$, proving that this ``universal behavior'' extends beyond the Szeg\H{o} class. The asymptotics of $E_n$ has also a curious interpretation in terms of the mutual energy of two relevant sequences of measures associated with $p_n$'s

Electrostatic models for zeros of polynomials: Old, new, and some open problems

15 pages, 2 figures.-- MSC2000 codes: Primary, 30C15; Secondary, 34C10; 33C45; 42C05; 82B23.MR#: MR2345246 (2008h:33017)Zbl#: Zbl 1131.30002We give a survey concerning both very classical and recent results on the electrostatic interpretation of the zeros of some well-known families of polynomials, and the interplay between these models and the asymptotic distribution of their zeros when the degree of the polynomials tends to infinity. The leading role is played by the differential equation satisfied by these polynomials. Some new developments, applications and open problems are presented.This research was supported, in part, by a grant of Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain, project code BFM2003-06335-C03-02 (FM), a grant from the Ministry of Education and Science of Spain, project code MTM2005-08648-C02-01 (AMF, PMG), by Junta de Andalucía, Grupo de Investigación FQM 0229 (AMF, PMG), by “Research Network on Constructive Complex Approximation (NeCCA)”, INTAS 03-51-6637 (FM, AMF), and by NATO Collaborative Linkage Grant “Orthogonal Polynomials: Theory, Applications and Generalizations”, ref. PST.CLG.979738 (FM, AMF).Publicad

Jacqueline Harpman's transgressive dystopian fantastic in ‘moi qui n'ai pas connu les hommes’:Between familiar territory and unknown worlds

Type I Hermite-Pad\'e polynomials for set of functions $f_0, f_1,..., f_s$ at infinity, $(Q_{n,0}f_0+Q_{n,1}f_1+Q_{n,2}f_2+...+Q_{n,s}f_s)(z)=O(\frac{1}{z^{sn+s}}), z\rightarrow \infty$ with the degree of all $Q_{n,k}<=n$. We describe an approach for finding the asymptotic zero distribution of these polynomials as $n\rightarrow \infty$ under the assumption that all $f'_js$ are semiclassical, i.e. their logarithmic derivatives are rational functions. In this situation $R_n$ and $Q_{n,k}f_k$ satisfy the same differential equation with polynomials coefficients. We discuss in more detail the case when $f'_k$s are powers of the same function $f (f_k=f^k)$; for illustration, the simplest non trivial situation of $s=2$ and $f$ having two branch points is analyzed in depth. Under these conditions, the ratio or comparative asymptotics of these polynomials is also discussed. From methodological considerations and in order to make the situation clearer, we start our exposition with the better known case of Pad\'e approximants (when $s=1$)

Strong asymptotics for Gegenbauer-Sobolev orthogonal polynomials

6 pages, no figures.-- MSC1991 codes: 33C25; 42CO5.Zbl#: Zbl 0895.33003We study the asymptotic behaviour of the monic orthogonal polynomials with respect to the Gegenbauer-Sobolev inner product (f,g)_S= + λ where = \int_{-1}\sp 1f(x)g(x)(1-x\sp 2)\sp {\alpha-\frac{1}{2}}dx, with α > -1/2 and λ > 0. The asymptotics of the zeros and norms of these polynomials are also established.Research by first (A.M.F.) and second (J.J.M.B.) was partially supported by Junta de Andalucía, Grupo de Investigación FQM 0229.Publicad