Related papers: From Heun Class Equations to Painlev\'e Equations
The Li\'enard equation is used in various applications. Therefore, constructing general analytical solutions of this equation is an important problem. Here we study connections between the Li\'enard equation and some equations from the…
In this paper, we provide a comprehensive method for constructing Lax pairs of discrete Painlev\'e equations by using a reduced hypercube structure. In particular, we consider the $A_5^{(1)}$-surface $q$-Painlev\'e system which has the…
In a recent work difference equations (Laguerre-Freud equations) for the bi-orthogonal polynomials and related quantities corresponding to the weight on the unit circle $ w(z)=\prod^m_{j=1}(z-z_j(t))^{\rho_j} $ were derived.Here it is shown…
We prove the meromorphy of solutions for a wide class of ordinary differential equations. These equations are given by invariant manifolds of non-linear partial differential equations integrable by the inverse scattering method. Some higher…
The Riemann-Hilbert approach for the equations ${\rm PIII(D_6)}$ and ${\rm PIII(D_7)}$ is studied in detail, involving moduli spaces for connections and monodromy data, Okamoto-Painlev\'e varieties, the Painlev\'e property, special…
An explicit form of the Lax pair for the q-difference Painleve equation with affine Weyl group symmetry of type E^{(1)}_8 is obtained. Its degeneration to E^{(1)}_7, E^{(1)}_6 and D^{(1)}_5 cases are also given.
Painleve equations belong to the class y'' + a_1 {y'}^3 + 3 a_2 {y'}^2 + 3 a_3 y' + a_4 = 0, where a_i=a_i(x,y). This class of equations is invariant under the general point transformation x=Phi(X,Y), y=Psi(X,Y) and it is therefore very…
A study is presented of superintegrable quantum systems in two-dimensional Euclidean space $E_2$ allowing the separation of variables in Cartesian coordinates. In addition to the Hamiltonian $H$ and the second order integral of motion $X$,…
Sch\"afke and Schmidt established that the asymptotics of the coefficients of the local solution to some linear differential equation is related to global structures of solutions. The Heun class equations have the accessory parameters, and…
We study the recurrence coefficients of the monic polynomials $P_n(z)$ orthogonal with respect to the deformed (also called semi-classical) Freud weight \begin{equation*} w_{\alpha}(x;s,N)=|x|^{\alpha}{\rm…
We study second order and third order linear differential equations with analytic coefficients under the viewpoint of finding formal solutions and studying their convergence. We address some untouched aspects of Frobenius methods for second…
It is well known that the sixth Painlev\'e equation $\PVI$ admits a group of B\"acklund transformations which is isomorphic to the affine Weyl group of type $\mathrm{D}_4^{(1)}$. Although various aspects of this unexpectedly large symmetry…
Linearization of coupled second order nonlinear ordinary differential equations (SNODEs) is one of the open and challenging problems in the theory of differential equations. In this paper we describe a simple and straightforward method to…
We give a classification for the small-$\tau$ asymptotic behaviours of solutions to the degenerate third Painlev\'e equation, $u^{''}(\tau) = \frac{(u^{\prime}(\tau))^{2}}{u(\tau)} - \frac{u^{\prime}(\tau)}{\tau} + \frac{1}{\tau}\left(-8…
We revisit several known versions of the Dehn--Sommerville relations in the context of: homology manifolds, semi-Eulerian complexes, general simplicial complexes, balanced semi-Eulerian complexes and general completely balanced complexes.…
We classify all functions satisfying non-trivial families of PVI equations. It turns out that, up to an Okamoto equivalence, there are exactly four families parameterized by affine planes or lines. Each affine space is generated by points…
We suggest a systematic procedure for classifying partial differential equations invariant with respect to low dimensional Lie algebras. This procedure is a proper synthesis of the infinitesimal Lie's method, technique of equivalence…
The first five classical Painlev\'e equations are known to have solutions described by divergent asymptotic power series near infinity. Here we prove that such solutions also exist for the infinite hierarchy of equations associated with the…
Using the only admissible rank-two realisations of the Lie algebra of the affine group in one dimension in terms of the Lie algebra of Lie symmetries of the Ermakov-Pinney (EP) equation, some classes of second order nonlinear ordinary…
The 'relativistic' Heun equation is an 8-coupling difference equation that generalizes the 4-coupling Heun differential equation. It can be viewed as the time-independent Schr\"odinger equation for an analytic difference operator introduced…