A high order qq-difference equation for qq-Hahn multiple orthogonal polynomials

A high order linear $q$-difference equation with polynomial coefficients having $q$-Hahn multiple orthogonal polynomials as eigenfunctions is given. The order of the equation is related to the number of orthogonality conditions that these polynomials satisfy. Some limiting situations when $q\to1$ are studied. Indeed, the difference equation for Hahn multiple orthogonal polynomials given in \cite{Lee} is corrected and obtained as a limiting case

On the speed of convergence to stationarity of the Erlang loss system

We consider the Erlang loss system, characterized by $N$ servers, Poisson arrivals and exponential service times, and allow the arrival rate to be a function of $N.$ We discuss representations and bounds for the rate of convergence to stationarity of the number of customers in the system, and display some bounds for the total variation distance between the time-dependent and stationary distributions. We also pay attention to time-dependent rates

Holography, Pade Approximants and Deconstruction

We investigate the relation between holographic calculations in 5D and the Migdal approach to correlation functions in large N theories. The latter employs Pade approximation to extrapolate short distance correlation functions to large distances. We make the Migdal/5D relation more precise by quantifying the correspondence between Pade approximation and the background and boundary conditions in 5D. We also establish a connection between the Migdal approach and the models of deconstructed dimensions.Comment: 28 page

qq-Classical orthogonal polynomials: A general difference calculus approach

It is well known that the classical families of orthogonal polynomials are characterized as eigenfunctions of a second order linear differential/difference operator. In this paper we present a study of classical orthogonal polynomials in a more general context by using the differential (or difference) calculus and Operator Theory. In such a way we obtain a unified representation of them. Furthermore, some well known results related to the Rodrigues operator are deduced. A more general characterization Theorem that the one given in [1] and [2] for the q-polynomials of the q-Askey and Hahn Tableaux, respectively, is established. Finally, the families of Askey-Wilson polynomials, q-Racah polynomials, Al-Salam & Carlitz I and II, and q-Meixner are considered. [1] R. Alvarez-Nodarse. On characterization of classical polynomials. J. Comput. Appl. Math., 196:320{337, 2006. [2] M. Alfaro and R. Alvarez-Nodarse. A characterization of the classical orthogonal discrete and q-polynomials. J. Comput. Appl. Math., 2006. In press.Comment: 18 page

Quantum central limit theorem for continuous-time quantum walks on odd graphs in quantum probability theory

The method of the quantum probability theory only requires simple structural data of graph and allows us to avoid a heavy combinational argument often necessary to obtain full description of spectrum of the adjacency matrix. In the present paper, by using the idea of calculation of the probability amplitudes for continuous-time quantum walk in terms of the quantum probability theory, we investigate quantum central limit theorem for continuous-time quantum walks on odd graphs.Comment: 19 page, 1 figure

Zeros of Orthogonal Polynomials Generated by the Geronimus Perturbation of Measures

Proceedings of: 14th International Conference Computational Science and Its Applications (ICCSA 2014). Guimarães, Portugal, June 30 – July 3, 2014This paper deals with monic orthogonal polynomial sequences (MOPS in short) generated by a Geronimus canonical spectral transformation of a positive Borel measure μ, i.e., (x−c) −1dμ(x)+Nδ(x−c), for some free parameter N ∈ IR+ and shift c. We analyze the behavior of the corresponding MOPS. In particular, we obtain such a behavior when the mass N tends to infinity as well as we characterize the precise values of N such the smallest (respectively, the largest) zero of these MOPS is located outside the support of the original measure μ. When μ is semi-classical, we obtain the ladder operators and the second order linear differential equation satisfied by the Geronimus perturbed MOPS, and we also give an electrostatic interpretation of the zero distribution in terms of a logarithmic potential interaction under the action of an external field. We analyze such an equilibrium problem when the mass point of the perturbation c is located outside the support of μ

Semigroups of distributions with linear Jacobi parameters

We show that a convolution semigroup of measures has Jacobi parameters polynomial in the convolution parameter $t$ if and only if the measures come from the Meixner class. Moreover, we prove the parallel result, in a more explicit way, for the free convolution and the free Meixner class. We then construct the class of measures satisfying the same property for the two-state free convolution. This class of two-state free convolution semigroups has not been considered explicitly before. We show that it also has Meixner-type properties. Specifically, it contains the analogs of the normal, Poisson, and binomial distributions, has a Laha-Lukacs-type characterization, and is related to the $q=0$ case of quadratic harnesses.Comment: v3: the article is merged back together with arXiv:1003.4025. A significant revision following suggestions by the referee. 2 pdf figure

Variational Mean Field approach to the Double Exchange Model

It has been recently shown that the double exchange Hamiltonian, with weak antiferromagnetic interactions, has a richer variety of first and second order transitions than previously anticipated, and that such transitions are consistent with the magnetic properties of manganites. Here we present a thorough discussion of the variational Mean Field approach that leads to the these results. We also show that the effect of the Berry phase turns out to be crucial to produce first order Paramagnetic-Ferromagnetic transitions near half filling with transition temperatures compatible with the experimental situation. The computation relies on two crucial facts: the use of a Mean Field ansatz that retains the complexity of a system of electrons with off-diagonal disorder, not fully taken into account by the Mean Field techniques, and the small but significant antiferromagnetic superexchange interaction between the localized spins.Comment: 13 pages, 11 postscript figures, revte

Gauge-Invariant Quasi-Free States on the Algebra of the Anyon Commutation Relations

Let $X=\mathbb R^2$ and let $q\in\mathbb C$, $|q|=1$. For $x=(x^1,x^2)$ and $y=(y^1,y^2)$ from $X^2$, we define a function $Q(x,y)$ to be equal to $q$ if $x^1y^1$, and to $\Re q$ if $x^1=y^1$. Let $\partial_x^+$, $\partial_x^-$ ($x\in X$) be operator-valued distributions such that $\partial_x^+$ is the adjoint of $\partial_x^-$. We say that $\partial_x^+$, $\partial_x^-$ satisfy the anyon commutation relations (ACR) if $\partial^+_x\partial_y^+=Q(y,x)\partial_y^+\partial_x^+$ for $x\ne y$ and $\partial^-_x\partial_y^+=\delta(x-y)+Q(x,y)\partial_y^+\partial^-_x$ for $(x,y)\in X^2$. In particular, for $q=1$, the ACR become the canonical commutation relations and for $q=-1$, the ACR become the canonical anticommutation relations. We define the ACR algebra as the algebra generated by operator-valued integrals of $\partial_x^+$, $\partial_x^-$. We construct a class of gauge-invariant quasi-free states on the ACR algebra. Each state from this class is completely determined by a positive self-adjoint operator $T$ on the real space $L^2(X,dx)$ which commutes with any operator of multiplication by a bounded function $\psi(x^1)$. In the case $\Re q0$), we discuss the corresponding particle density $\rho(x):=\partial_x^+\partial_x^-$. For $\Re q\in(0,1]$, using a renormalization, we rigorously define a vacuum state on the commutative algebra generated by operator-valued integrals of $\rho(x)$. This state is given by a negative binomial point process. A scaling limit of these states as $\kappa\to\infty$ gives the gamma random measure, depending on parameter $\Re q$