A family of Nikishin systems with periodic recurrence coefficients

Suppose we have a Nikishin system of $p$ measures with the $k$th generating measure of the Nikishin system supported on an interval \Delta_k\subset\er with $\Delta_k\cap\Delta_{k+1}=\emptyset$ for all $k$. It is well known that the corresponding staircase sequence of multiple orthogonal polynomials satisfies a $(p+2)$-term recurrence relation whose recurrence coefficients, under appropriate assumptions on the generating measures, have periodic limits of period $p$. (The limit values depend only on the positions of the intervals $\Delta_k$.) Taking these periodic limit values as the coefficients of a new $(p+2)$-term recurrence relation, we construct a canonical sequence of monic polynomials $\{P_{n}\}_{n=0}^{\infty}$, the so-called \emph{Chebyshev-Nikishin polynomials}. We show that the polynomials $P_{n}$ themselves form a sequence of multiple orthogonal polynomials with respect to some Nikishin system of measures, with the $k$th generating measure being absolutely continuous on $\Delta_{k}$. In this way we generalize a result of the third author and Rocha \cite{LopRoc} for the case $p=2$. The proof uses the connection with block Toeplitz matrices, and with a certain Riemann surface of genus zero. We also obtain strong asymptotics and an exact Widom-type formula for the second kind functions of the Nikishin system for $\{P_{n}\}_{n=0}^{\infty}$.Comment: 30 pages, minor change

Logarithmic potential of Hermite polynomials and information entropies of the harmonic oscillator eigenstates

13 pages, 1 figure.-- PACS nrs.: 03.65.Ge, 02.10.Nj, 02.10.Sp.MR#: MR1471913 (99c:81031)Zbl#: Zbl 0891.33007The problem of calculating the information entropy in both position and momentum spaces for the nth stationary state of the one-dimensional quantum harmonic oscillator reduces to the evaluation of the logarithmic potential V_n(t)=-\int e\sp {-x\sp 2}H_n\sp 2(x)\log -t x at the zeros of the Hermite polynomial Hn(x). Here, a closed analytical expression for Vn(t) is obtained, which in turn yields an exact analytical expression for the entropies when the exact location of the zeros of Hn(x) is known. An inequality for the values of Vn(t) at the zeros of Hn(x) is conjectured, which leads to a new, nonvariational, upper bound for the entropies. Finally, the exact formula for Vn(t) is written in an alternative way, which allows the entropies to be expressed in terms of the even-order spectral moments of the Hermite polynomials. The asymptotic (n>>1) limit of this alternative expression for the entropies is discussed, and the conjectured upper bound for the entropies is proved to be asymptotically validThe author gratefully acknowledges the financial support from the Fundació Aula (Barcelona, Spain).Publicad

Asymptotics for Hermite-Pade rational approximants for two analytic functions with separated pairs of branch points (case of genus 0)

We investigate the asymptotic behavior for type II Hermite-Pade approximation to two functions, where each function has two branch points and the pairs of branch points are separated. We give a classification of the cases such that the limiting counting measures for the poles of the Hermite-Pade approximants are described by an algebraic function of order 3 and genus 0. This situation gives rise to a vector-potential equilibrium problem for three measures and the poles of the common denominator are asymptotically distributed like one of these measures. We also work out the strong asymptotics for the corresponding Hermite-Pade approximants by using a 3x3 Riemann-Hilbert problem that characterizes this Hermite-Pade approximation problem.Comment: 102 pages, 31 figure

Asymptotics for multiple Meixner polynomials

. We use an algebraic function formulation for the solution of the equilibrium problem with constraint to describe their zero distribution. Moreover, analyzing the limiting behavior of the coefficients of the recurrence relations for Multiple Meixner polynomials we obtain the main term of their asymptotics. (C) 2013 Elsevier Inc. All rights reserved.The research of J. Arvesú was partially supported by the research grant MTM2012-36732-C03-01 of the Ministerio de Educación y Ciencia of Spain and grants CCG07-UC3M/ESP-3339 and CCG08-UC3M/ESP-4516 from Comunidad Autónoma de Madrid