Asymptotics of multiple orthogonal polynomials for a system of two measures supported on a starlike set

For a system of two measures supported on a starlike set in the complex plane, we study asymptotic properties of associated multiple orthogonal polynomials $Q_{n}$ and their recurrence coefficients. These measures are assumed to form a Nikishin-type system, and the polynomials $Q_{n}$ satisfy a three-term recurrence relation of order three with positive coefficients. Under certain assumptions on the orthogonality measures, we prove that the sequence of ratios $\{Q_{n+1}/Q_{n}\}$ has four different periodic limits, and we describe these limits in terms of a conformal representation of a compact Riemann surface. Several relations are found involving these limiting functions and the limiting values of the recurrence coefficients. We also study the $n$th root asymptotic behavior and zero asymptotic distribution of $Q_{n}$.Comment: 31 page

Relative Asymptotic of Multiple Orthogonal Polynomials for Nikishin Systems

We prove relative asymptotic for the ratio of two sequences of multiple orthogonal polynomials with respect to Nikishin system of measures. The first Nikishin system ${\mathcal{N}}(\sigma_1,...,\sigma_m)$ is such that for each $k$, $\sigma_k$ has constant sign on its compact support \supp {\sigma_k} \subset \mathbb{R} consisting of an interval $\widetilde{\Delta}_k$, on which $|\sigma_k^{\prime}| > 0$ almost everywhere, and a discrete set without accumulation points in $\mathbb{R} \setminus \widetilde{\Delta}_k$. If {Co}(\supp {\sigma_k}) = \Delta_k denotes the smallest interval containing \supp {\sigma_k}, we assume that $\Delta_k \cap \Delta_{k+1} = \emptyset$, $k=1,...,m-1$. The second Nikishin system ${\mathcal{N}}(r_1\sigma_1,...,r_m\sigma_m)$ is a perturbation of the first by means of rational functions $r_k$, $k=1,...,m,$ whose zeros and poles lie in $\mathbb{C} \setminus \cup_{k=1}^m \Delta_k$.Comment: 30 page

High order three-term recursions, Riemann-Hilbert minors and Nikishin systems on star-like sets

We study monic polynomials $Q_n(x)$ generated by a high order three-term recursion $xQ_n(x)=Q_{n+1}(x)+a_{n-p} Q_{n-p}(x)$ with arbitrary $p\geq 1$ and $a_n>0$ for all $n$. The recursion is encoded by a two-diagonal Hessenberg operator $H$. One of our main results is that, for periodic coefficients $a_n$ and under certain conditions, the $Q_n$ are multiple orthogonal polynomials with respect to a Nikishin system of orthogonality measures supported on star-like sets in the complex plane. This improves a recent result of Aptekarev-Kalyagin-Saff where a formal connection with Nikishin systems was obtained in the case when $\sum_{n=0}^{\infty}|a_n-a|0$. An important tool in this paper is the study of "Riemann-Hilbert minors", or equivalently, the "generalized eigenvalues" of the Hessenberg matrix $H$. We prove interlacing relations for the generalized eigenvalues by using totally positive matrices. In the case of asymptotically periodic coefficients $a_n$, we find weak and ratio asymptotics for the Riemann-Hilbert minors and we obtain a connection with a vector equilibrium problem. We anticipate that in the future, the study of Riemann-Hilbert minors may prove useful for more general classes of multiple orthogonal polynomials.Comment: 59 pages, 3 figure

Nikishin systems on star-like sets: Ratio asymptotics of the associated multiple orthogonal polynomials, II

In this paper we continue the investigations initiated in \cite{LopLopstar} on ratio asymptotics of multiple orthogonal polynomials and functions of the second kind associated with Nikishin systems on star-like sets. We describe in detail the limiting functions found in \cite{LopLopstar}, expressing them in terms of certain conformal mappings defined on a compact Riemann surface of genus zero. We also express the limiting values of the recurrence coefficients, which are shown to be strictly positive, in terms of certain values of the conformal mappings. As a consequence, the limits depend exclusively on the location of the intervals determined by the supports of the measures that generate the Nikishin system.Comment: Change in title, corrections have been made. 27 page

Ratio asymptotic of Hermite-Padé orthogonal polynomials for Nikishin systems. II

26 pages, no figures.-- MSC2000 codes: Primary 42C05, 30E10; Secondary 41A21.MR#: MR2419380 (2009e:42053)Zbl#: Zbl 1153.42013We prove ratio asymptotic for sequences of multiple orthogonal polynomials with respect to a Nikishin system of measures ${\cal N}(\sigma_1,\dots,\sigma_m)$ such that for each k, σ_k has constant sign on its support consisting on an interval $\tilde\Delta_k$, on which $sigma_k'>0$ almost everywhere, and a set without accumulation points in $\Bbb R\setminus \tilde\Delta_k$.Both authors received support from grants MTM 2006-13000-C03-02 of Ministerio de Ciencia y Tecnología, UC3M-CAM MTM-05-033, and UC3M-CAM CCG-06003M/ESP-0690 of Comunidad Autónoma de Madrid.Publicad

Asymptotics of the optimal values of potentials generated by greedy energy sequences on the unit circle

For the Riesz and logarithmic potentials, we consider greedy energy sequences $(a_n)_{n=0}^\infty$ on the unit circle $S^1$, constructed in such a way that for every $n\geq 1$, the discrete potential generated by the first $n$ points $a_0,\ldots,a_{n-1}$ of the sequence attains its minimum (say $U_n$) at $a_n$. We obtain asymptotic formulae that describe the behavior of $U_n$ as $n\to\infty$, in terms of certain bounded arithmetic functions with a doubling periodicity property. As previously shown in \cite{LopMc2}, after properly translating and scaling $U_n$, one obtains a new sequence $(F_n)$ that is bounded and divergent. We find the exact value of $\liminf F_n$ (the value of $\limsup F_n$ was already given in \cite{LopMc2}), and show that the interval $[\liminf F_n,\limsup F_n]$ comprises all the limit points of the sequence $(F_n)$.Comment: 20 page