112 research outputs found
Rate of Relative Growth of Orthogonal Polynomials
AbstractLet 0 < p < â and Ï”n = |an â a/2| + |bn â b| â 0. define a polynomial sequence {pn} by the recurrence relation xpn(x) = an + 1(x) + bnpn(x) + anpn â 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 Ï”n. 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 where is the integral
operator with kernel 1/\pi} {\sin\pi(x-y)\over x-y} \chi_I(y). Here
and is the characteristic function
of the set . In the Gaussian Unitary Ensemble (GUE) the probability that no
eigenvalues lie in is equal to . Also is a tau-function
and we present a new simplified derivation of the system of nonlinear
completely integrable equations (the '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 we give an asymptotic formula for , which is
the probability in the GUE that exactly eigenvalues lie in an interval of
length .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
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