439 research outputs found
Spectral methods for bivariate Markov processes with diffusion and discrete components and a variant of the Wright-Fisher model
The aim of this paper is to study differential and spectral properties of the
infinitesimal operator of two dimensional Markov processes with diffusion and
discrete components. The infinitesimal operator is now a second-order
differential operator with matrix-valued coefficients, from which we can derive
backward and forward equations, a spectral representation of the probability
density, study recurrence of the process and the corresponding invariant
distribution. All these results are applied to an example coming from group
representation theory which can be viewed as a variant of the Wright-Fisher
model involving only mutation effects.Comment: 6 figure
Some examples of matrix-valued orthogonal functions having a differential and an integral operator as eigenfunctions
The aim of this paper is to show some examples of matrix-valued orthogonal
functions on the real line which are simultaneously eigenfunctions of a
second-order differential operator of Schr\"{o}dinger type and an integral
operator of Fourier type. As a consequence we derive integral representations
of these functions as well as other useful structural formulas. Some of these
functions are plotted to show the relationship with the Hermite or wave
functions
Non-commutative Painleve' equations and Hermite-type matrix orthogonal polynomials
We study double integral representations of Christoffel-Darboux kernels
associated with two examples of Hermite-type matrix orthogonal polynomials. We
show that the Fredholm determinants connected with these kernels are related
through the Its-Izergin-Korepin-Slavnov (IIKS) theory with a certain
Riemann-Hilbert problem. Using this Riemann-Hilbert problem we obtain a Lax
pair whose compatibility conditions lead to a non-commutative version of the
Painleve' IV differential equation for each family.Comment: Final version, accepted for publication on CMP: Communications in
Mathematical Physics. 24 pages, 1 figur
Constructing Krall-Hahn orthogonal polynomials
Given a sequence of polynomials , an algebra of operators acting in the linear space of polynomials and an operator with , where is any arbitrary eigenvalue,
we construct a new sequence of polynomials by considering a linear
combination of consecutive :
. Using the concept of
-operator, we determine the structure of the sequences
in order that the polynomials are
eigenfunctions of an operator in the algebra . As an application,
from the classical discrete family of Hahn polynomials we construct orthogonal
polynomials which are also eigenfunctions of higher-order difference
operators.Comment: 26 pages. arXiv admin note: text overlap with arXiv:1307.1326,
arXiv:1407.697
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