1,698 research outputs found
Semi-classical Orthogonal Polynomial Systems on Non-uniform Lattices, Deformations of the Askey Table and Analogs of Isomonodromy
A -semi-classical weight is one which satisfies a particular
linear, first order homogeneous equation in a divided-difference operator
. It is known that the system of polynomials, orthogonal with
respect to this weight, and the associated functions satisfy a linear, first
order homogeneous matrix equation in the divided-difference operator termed the
spectral equation. Attached to the spectral equation is a structure which
constitutes a number of relations such as those arising from compatibility with
the three-term recurrence relation. Here this structure is elucidated in the
general case of quadratic lattices. The simplest examples of the
-semi-classical orthogonal polynomial systems are precisely those
in the Askey table of hypergeometric and basic hypergeometric orthogonal
polynomials. However within the -semi-classical class it is
entirely natural to define a generalisation of the Askey table weights which
involve a deformation with respect to new deformation variables. We completely
construct the analogous structures arising from such deformations and their
relations with the other elements of the theory. As an example we treat the
first non-trivial deformation of the Askey-Wilson orthogonal polynomial system
defined by the -quadratic divided-difference operator, the Askey-Wilson
operator, and derive the coupled first order divided-difference equations
characterising its evolution in the deformation variable. We show that this
system is a member of a sequence of classical solutions to the
-Painlev\'e system.Comment: Submitted to Duke Mathematical Journal on 5th April 201
The Distribution of the first Eigenvalue Spacing at the Hard Edge of the Laguerre Unitary Ensemble
The distribution function for the first eigenvalue spacing in the Laguerre
unitary ensemble of finite rank random matrices is found in terms of a
Painlev\'e V system, and the solution of its associated linear isomonodromic
system. In particular it is characterised by the polynomial solutions to the
isomonodromic equations which are also orthogonal with respect to a deformation
of the Laguerre weight. In the scaling to the hard edge regime we find an
analogous situation where a certain Painlev\'e \IIId system and its associated
linear isomonodromic system characterise the scaled distribution. We undertake
extensive analytical studies of this system and use this knowledge to
accurately compute the distribution and its moments for various values of the
parameter . In particular choosing allows the first
eigenvalue spacing distribution for random real orthogonal matrices to be
computed.Comment: 65 pages, 1 eps figure, typos and references correcte
Bi-orthogonal systems on the unit circle, Regular Semi-Classical Weights and Integrable Systems - II
We derive the Christoffel-Geronimus-Uvarov transformations of a system of
bi-orthogonal polynomials and associated functions on the unit circle, that is
to say the modification of the system corresponding to a rational modification
of the weight function. In the specialisation of the weight function to the
regular semi-classical case with an arbitrary number of regular singularities the bi-orthogonal system is known to be isomonodromy
preserving with respect to deformations of the singular points. If the zeros
and poles of the Christoffel-Geronimus-Uvarov factors coincide with the
singularities then we have the Schlesinger transformations of this
isomonodromic system. Compatibility of the Schlesinger transformations with the
other structures of the system - the recurrence relations, the spectral
derivatives and deformation derivatives is explicitly deduced. Various forms of
Hirota-Miwa equations are derived for the -functions or equivalently
Toeplitz determinants of the system.Comment: to appear J. Approx. Theor
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