Related papers: Differential equations for deformed Laguerre polyn…
This note supplements the work of Gomez-Ullate, Kamran and Milson on the X_(1)-Laguerre polynomials which are orthogonal in a weighted Hilbert function space on the positive half-line of the real line. These polynomials are generated by a…
A scale of the Frechet spaces of exponential type entire functions of one complex variable is considered. Certain special properties of subsets of these spaces consisting of Laguerre entire functions, which are obtained as uniform limits on…
The spectral density for random matrix $\beta$ ensembles can be written in terms of the average of the absolute value of the characteristic polynomial raised to the power of $\beta$, which for even $\beta$ is a polynomial of degree…
Infinitely many explicit solutions of certain second-order differential equations with an apparent singularity of characteristic exponent -2 are constructed by adjusting the parameter of the multi-indexed Laguerre polynomials.
We consider the problem of the exact computation of the marginal eigenvalue distributions in the Laguerre and Jacobi $\beta$ ensembles. In the case $\beta=1$ this is a question of long standing in the mathematical statistics literature. A…
Uniform asymptotic expansions involving exponential and Airy functions are obtained for Laguerre polynomials $L_{n}^{(\alpha)}(x)$, as well as complementary confluent hypergeometric functions. The expansions are valid for $n$ large and…
In this paper, we present a stable and efficient approach for constructing Laguerre pseudospectral differentiation matrices. The proposed method reformulates the off-diagonal entries and computes all required quantities simultaneously using…
The level spacing distributions which arise when one rescales the Laguerre or Jacobi ensembles of hermitian matrices is studied. These distributions are expressible in terms of a Fredholm determinant of an integral operator whose kernel is…
The Laguerre symmetric functions introduced in the note are indexed by arbitrary partitions and depend on two continuous parameters. The top degree homogeneous component of every Laguerre symmetric function coincides with the Schur function…
The focus of this paper is on the distribution function of the rightmost eigenvalue for the complex elliptic Ginibre ensemble in the limit of weak non-Hermiticity. We show how the limiting distribution function can be expressed in terms of…
We will discuss how we can obtain new quantum superintegrable Hamiltonians allowing the separation of variables in Cartesian coordinates with higher order integrals of motion from ladder operators. We will discuss also how higher order…
The Laplace transform is a useful and powerful analytic tool with applications to several areas of applied mathematics, including differential equations, probability and statistics. Similarly to the inversion of the Fourier transform,…
Unstable separatrix solutions for the first and second Painlev\'e transcendents are studied both numerically and analytically. For a fixed initial condition, say $y(0)=0$, there is a discrete set of initial slopes $y'(0)=b_n$ that give rise…
We study a family of Laguerre--Sobolev orthogonal polynomials associated with a Sobolev inner product arising from second--order boundary value problems on the semi--infinite interval $(0,+\infty)$. These polynomials generate an orthogonal…
The Painlev\'{e} equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of ``classical" weights multiplied by suitable ``deformation factors", usually dependent on a…
It is well-known that differential Painlev\'e equations can be written in a Hamiltonian form. However, a coordinate form of such representation is far from unique -- there are many very different Hamiltonians that result in the same…
We derive the Lagrangians of the higher-order Painlev\'e equations using Jacobi's last multiplier technique. Some of these higher-order differential equations display certain remarkable properties like passing the Painlev\'e test and…
We present two complementary methods, each applicable in a different range, to evaluate the distribution of the lowest eigenvalue of random matrices in a Jacobi ensemble. The first method solves an associated Painleve VI nonlinear…
We study the bispectrality of Laguerre type polynomials, which are defined by taking suitable linear combinations of a fixed number of consecutive Laguerre polynomials. These Laguerre type polynomials are eigenfunctions of higher-order…
In this paper, we introduce the Laguerre bounded variation space and the Laguerre perimeter, thereby investigating their properties. Moreover, we prove the isoperimetric inequality and the Sobolev inequality in the Laguerre setting. As…