321 research outputs found
Multiplier Sequences for Simple Sets of Polynomials
In this paper we give a new characterization of simple sets of polynomials B
with the property that the set of B-multiplier sequences contains all
Q-multiplier sequence for every simple set Q. We characterize sequences of real
numbers which are multiplier sequences for every simple set Q, and obtain some
results toward the partitioning of the set of classical multiplier sequences
Polynomial solutions of nonlinear integral equations
We analyze the polynomial solutions of a nonlinear integral equation,
generalizing the work of C. Bender and E. Ben-Naim. We show that, in some
cases, an orthogonal solution exists and we give its general form in terms of
kernel polynomials.Comment: 10 page
Phase transition in a log-normal Markov functional model
We derive the exact solution of a one-dimensional Markov functional model
with log-normally distributed interest rates in discrete time. The model is
shown to have two distinct limiting states, corresponding to small and
asymptotically large volatilities, respectively. These volatility regimes are
separated by a phase transition at some critical value of the volatility. We
investigate the conditions under which this phase transition occurs, and show
that it is related to the position of the zeros of an appropriately defined
generating function in the complex plane, in analogy with the Lee-Yang theory
of the phase transitions in condensed matter physics.Comment: 9 pages, 5 figures. v2: Added asymptotic expressions for the
convexity-adjusted Libors in the small and large volatility limits. v3: Added
one reference. Final version to appear in Journal of Mathematical Physic
Congruences concerning Jacobi polynomials and Ap\'ery-like formulae
Let be a prime. We prove congruences modulo for sums of the
general form and
with . We also consider the
special case of the former sum, where the congruences hold
modulo .Comment: to appear in Int. J. Number Theor
Asymptotic corrections to the eigenvalue density of the GUE and LUE
We obtain correction terms to the large N asymptotic expansions of the
eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of
random N by N matrices, both in the bulk of the spectrum and near the spectral
edge. This is achieved by using the well known orthogonal polynomial expression
for the kernel to construct a double contour integral representation for the
density, to which we apply the saddle point method. The main correction to the
bulk density is oscillatory in N and depends on the distribution function of
the limiting density, while the corrections to the Airy kernel at the soft edge
are again expressed in terms of the Airy function and its first derivative. We
demonstrate numerically that these expansions are very accurate. A matching is
exhibited between the asymptotic expansion of the bulk density, expanded about
the edge, and the asymptotic expansion of the edge density, expanded into the
bulk.Comment: 14 pages, 4 figure
Explicit Integration of the Full Symmetric Toda Hierarchy and the Sorting Property
We give an explicit formula for the solution to the initial value problem of
the full symmetric Toda hierarchy. The formula is obtained by the
orthogonalization procedure of Szeg\"{o}, and is also interpreted as a
consequence of the QR factorization method of Symes \cite{symes}. The sorting
property of the dynamics is also proved for the case of a generic symmetric
matrix in the sense described in the text, and generalizations of tridiagonal
formulae are given for the case of matrices with nonzero diagonals.Comment: 13 pages, Latex
Chern-Simons matrix models and Stieltjes-Wigert polynomials
Employing the random matrix formulation of Chern-Simons theory on Seifert
manifolds, we show how the Stieltjes-Wigert orthogonal polynomials are useful
in exact computations in Chern-Simons matrix models. We construct a
biorthogonal extension of the Stieltjes-Wigert polynomials, not available in
the literature, necessary to study Chern-Simons matrix models when the geometry
is a lens space. We also discuss several other results based on the properties
of the polynomials: the equivalence between the Stieltjes-Wigert matrix model
and the discrete model that appears in q-2D Yang-Mills and the relationship
with Rogers-Szego polynomials and the corresponding equivalence with an unitary
matrix model. Finally, we also give a detailed proof of a result that relates
quantum dimensions with averages of Schur polynomials in the Stieltjes-Wigert
ensemble.Comment: 25 pages, AMS-LaTe
Boundary conditions associated with the Painlev\'e III' and V evaluations of some random matrix averages
In a previous work a random matrix average for the Laguerre unitary ensemble,
generalising the generating function for the probability that an interval at the hard edge contains eigenvalues, was evaluated in terms of
a Painlev\'e V transcendent in -form. However the boundary conditions
for the corresponding differential equation were not specified for the full
parameter space. Here this task is accomplished in general, and the obtained
functional form is compared against the most general small behaviour of
the Painlev\'e V equation in -form known from the work of Jimbo. An
analogous study is carried out for the the hard edge scaling limit of the
random matrix average, which we have previously evaluated in terms of a
Painlev\'e \IIId transcendent in -form. An application of the latter
result is given to the rapid evaluation of a Hankel determinant appearing in a
recent work of Conrey, Rubinstein and Snaith relating to the derivative of the
Riemann zeta function
Edgeworth Expansion of the Largest Eigenvalue Distribution Function of GUE Revisited
We derive expansions of the resolvent
Rn(x;y;t)=(Qn(x;t)Pn(y;t)-Qn(y;t)Pn(x;t))/(x-y) of the Hermite kernel Kn at the
edge of the spectrum of the finite n Gaussian Unitary Ensemble (GUEn) and the
finite n expansion of Qn(x;t) and Pn(x;t). Using these large n expansions, we
give another proof of the derivation of an Edgeworth type theorem for the
largest eigenvalue distribution function of GUEn. We conclude with a brief
discussion on the derivation of the probability distribution function of the
corresponding largest eigenvalue in the Gaussian Orthogonal Ensemble (GOEn) and
Gaussian Symplectic Ensembles (GSEn)
The Heine-Stieltjes correspondence and the polynomial approach to the standard pairing problem
A new approach for solving the Bethe ansatz (Gaudin-Richardson) equations of
the standard pairing problem is established based on the Heine-Stieltjes
correspondence. For pairs of valence nucleons on different
single-particle levels, it is found that solutions of the Bethe ansatz
equations can be obtained from one (k+1)x(k+1) and one (n-1)x(k+1) matrices,
which are associated with the extended Heine-Stieltjes and Van Vleck
polynomials, respectively. Since the coefficients in these polynomials are free
from divergence with variations in contrast to the original Bethe ansatz
equations, the approach thus provides with a new efficient and systematic way
to solve the problem, which, by extension, can also be used to solve a large
class of Gaudin-type quantum many-body problems and to establish a new
efficient angular momentum projection method for multi-particle systems.Comment: ReVTeX, 4 pages, no figur
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