Rate of Relative Growth of Orthogonal Polynomials

AbstractLet 0 < p < ∞ and ϵn = |an − a/2| + |bn − b| → 0. define a polynomial sequence {pn} by the recurrence relation xpn(x) = an + 1(x) + bnpn(x) + anpn − 1(x). It was proved that [formula] This paper investigates the rate of growth of such a sequence relative to its sums in terms of the rate of convergance of ϵn. Some applications to orthogonal polynomials are given

Unique positive solution for an alternative discrete Painlevé I equation

We show that the alternative discrete Painleve I equation has a unique solution which remains positive for all n >0. Furthermore, we identify this positive solution in terms of a special solution of the second Painleve equation involving the Airy function Ai(t). The special-function solutions of the second Painleve equation involving only the Airy function Ai(t) therefore have the property that they remain positive for all n>0 and all t>0, which is a new characterization of these special solutions of the second Painlevé equation and the alternative discrete Painlevé I equation

Cosine and Sine Operators Related with Orthogonal Polynomial Sets on the Intervall [-1,1]

The quantization of phase is still an open problem. In the approach of Susskind and Glogower so called cosine and sine operators play a fundamental role. Their eigenstates in the Fock representation are related with the Chebyshev polynomials of the second kind. Here we introduce more general cosine and sine operators whose eigenfunctions in the Fock basis are related in a similar way with arbitrary orthogonal polynomial sets on the intervall [-1,1]. To each polynomial set defined in terms of a weight function there corresponds a pair of cosine and sine operators. Depending on the symmetry of the weight function we distinguish generalized or extended operators. Their eigenstates are used to define cosine and sine representations and probability distributions. We consider also the inverse arccosine and arcsine operators and use their eigenstates to define cosine-phase and sine-phase distributions, respectively. Specific, numerical and graphical results are given for the classical orthogonal polynomials and for particular Fock and coherent states.Comment: 1 tex-file (24 pages), 11 figure

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

Some Orthogonal Polynomials Arising from Coherent States

We explore in this paper some orthogonal polynomials which are naturally associated to certain families of coherent states, often referred to as nonlinear coherent states in the quantum optics literature. Some examples turn out to be known orthogonal polynomials but in many cases we encounter a general class of new orthogonal polynomials for which we establish several qualitative results.Comment: 21 page

Introduction to Random Matrices

These notes provide an introduction to the theory of random matrices. The central quantity studied is $\tau(a)= det(1-K)$ where $K$ is the integral operator with kernel 1/\pi} {\sin\pi(x-y)\over x-y} \chi_I(y). Here $I=\bigcup_j(a_{2j-1},a_{2j})$ and $\chi_I(y)$ is the characteristic function of the set $I$. In the Gaussian Unitary Ensemble (GUE) the probability that no eigenvalues lie in $I$ is equal to $\tau(a)$. Also $\tau(a)$ is a tau-function and we present a new simplified derivation of the system of nonlinear completely integrable equations (the $a_j$'s are the independent variables) that were first derived by Jimbo, Miwa, M{\^o}ri, and Sato in 1980. In the case of a single interval these equations are reducible to a Painlev{\'e} V equation. For large $s$ we give an asymptotic formula for $E_2(n;s)$, which is the probability in the GUE that exactly $n$ eigenvalues lie in an interval of length $s$.Comment: 44 page

Two-band random matrices

Spectral correlations in unitary invariant, non-Gaussian ensembles of large random matrices possessing an eigenvalue gap are studied within the framework of the orthogonal polynomial technique. Both local and global characteristics of spectra are directly reconstructed from the recurrence equation for orthogonal polynomials associated with a given random matrix ensemble. It is established that an eigenvalue gap does not affect the local eigenvalue correlations which follow the universal sine and the universal multicritical laws in the bulk and soft-edge scaling limits, respectively. By contrast, global smoothed eigenvalue correlations do reflect the presence of a gap, and are shown to satisfy a new universal law exhibiting a sharp dependence on the odd/even dimension of random matrices whose spectra are bounded. In the case of unbounded spectrum, the corresponding universal `density-density' correlator is conjectured to be generic for chaotic systems with a forbidden gap and broken time reversal symmetry.Comment: 12 pages (latex), references added, discussion enlarge

The impact of Stieltjes' work on continued fractions and orthogonal polynomials

Stieltjes' work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes' ideas and work in research done after his death, with an emphasis on the theory of orthogonal polynomials