Related papers: Painlev\'e IV: roots and zeros
The Lie point symmetries of ordinary differential equations (ODEs) that are candidates for having the Painlev\'e property are explored for ODEs of order $n =2, \dots ,5$. Among the 6 ODEs identifying the Painlev\'e transcendents only…
In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlev\'e equations, with a particular emphasis on the discrete Painlev\'e equations. The theory is controlled by the…
The purpose of this paper is to show how the problem of finding the zeros of unilateral n-order quaternionic polynomials can be solved by determining the eigen-vectors of the corresponding companion matrix. This approach, probably…
This is the last part of a series of three papers entitled "Four-dimensional Painlev\'e-type equations associated with ramified linear equations". In this series of papers we aim to construct the complete degeneration scheme of…
This paper presents new formulary solutions for quantic polynomial equations in general forms, where we present five solutions for any fifth degree polynomial equation with real coefficients, and thereby having the possibility to calculate…
In this paper we consider a fourth order equation involving the critical Sobolev exponent on a bounded and smooth domain in $\R^6$. Using theory of critical points at infinity, we give some topological conditions on a given function defined…
We solve the metrisability problem for the six Painlev\'e equations, and more generally for all 2nd order ODEs with Painlev\'e property, and determine for which of these equations their integral curves are geodesics of a (pseudo) Riemannian…
We consider the Clarkson-McLeod solutions of the fourth Painlev\'e equation. This family of solutions behave like $\kappa D_{\alpha-\frac{1}{2}}^2(\sqrt{2}x)$ as $x\rightarrow +\infty$, where $\kappa $ is an arbitrary real constant and…
We focus on Fuchsian equations with four accessory parameters and three singular points. We see that the Fuchsian equations admit a "degeneration scheme" in some sense, which is expected to give rise to a degeneration scheme of discrete…
The degenerate third Painleve' equation, $u"(t)=(u'(t))^2/u(t)-u'(t)/t+1/t(-8c u^2(t)+2ab)+b^2/u(t)$, where $c=+/-1$, $b>0$, and $a$ is a complex parameter, is studied via the Isomonodromy Deformation Method. Asymptotics of general regular…
In this paper, we derive explicit formulas for computing the roots of $ax^{2}+bx+c=0$ with $a$ being not invertible in split quaternion algebra. We also imitate the approach developed by Opfer, Janovska and Falcao etc. to verify our results…
We study polynomials that are orthogonal with respect to the modified Laguerre weight $z^{-n + \nu} e^{-Nz} (z-1)^{2b}$ in the limit where $n, N \to \infty$ with $N/n \to 1$ and $\nu$ is a fixed number in $\mathbb{R} \setminus…
Necessary discretization rules to preserve the Painlev\'e property are stated. A new method is added to the discrete Painlev\'e test, which perturbs the continuous limit and generates infinitely many no-log conditions.
Orthogonal polynomials and multiple orthogonal polynomials are interesting special functions because there is a beautiful theory for them, with many examples and useful applications in mathematical physics, numerical analysis, statistics…
In an earlier work, the authors have determined all possible weights $n$ for which there exists a vanishing sum $\zeta_1+\cdots +\zeta_n=0$ of $m$th roots of unity $\zeta_i$ in characteristic 0. In this paper, the same problem is studied in…
We offer elementary proofs for fundamental properties of solutions to the homogeneous second Painlev\'e equation.
The explicit integrability of second order ordinary differential equations invariant under time-translation and rescaling is investigated. Quadratic systems generated from the linearisable version of this class of equations are analysed to…
Under the quadratic-decay-conditions of the radial curvatures of an end, we shall derive growth estimates of solutions to the eigenvalue equation and show the absence of eigenvalues.
The existence of numerical solutions to a fourth order singular boundary value problem arising in the theory of epitaxial growth is studied. An iterative numerical method is applied on a second order nonlinear singular boundary value…
We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semi-classical Laguerre weight and classical solutions of the fourth Painlev\'e equation. We show that the coefficients in these…