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We tackle the regularisation of a differential system related to generalised Krawtchouk polynomials. We show a straightforward connection between certain auxiliary quantities involving the recurrence coefficients of these polynomials and…
A determinant formula for a class of algebraic solutions to Painlev\'e VI equation (P$_{\rm VI}$) is presented. This expression is regarded as a special case of the universal characters. The entries of the determinant are given by the…
Some properties and relations satisfied by the polynomial solutions of a bispectral problem are studied. Given a finite order differential operator, under certain restrictions, its polynomial eigenfunctions are explicitly obtained, as well…
Criteria are given for determining whether an irreducible sextic equation with rational coefficients is algebraically solvable over the complex numbers.
Although the solutions of Painlev\'e equations are transcendental in the sense that they cannot be expressed in terms of known elementary functions, there do exist rational solutions for specialized values of the equation parameters. A very…
We will classify all rational transformations which change the confluent hypergeometric equations to linear equations of the Painleve type from the first to the fifth. We show such rational transformations correspond to almost all of…
A determinant expression for the rational solutions of the Painlev\'e III (P$_{\rm III}$) equation whose entries are the Laguerre polynomials is given. Degeneration of this determinant expression to that for the rational solutions of…
We utilise a recent approach via the so-called re-scaling method to derive a unified and comprehensive theory of the solutions to Painleve's differential equations (I), (II) and (IV), with emphasis on the most elaborate equation (IV).
We give a new approach to the symmetries of the Painlev\'e equations $P_{V},P_{IV},P_{III}$ and $P_{II}$, respectively. Moreover, we make natural extensions to fourth-order analogues for each of the Painlev\'e equations $P_{V}$ and…
This paper presents new six solutions for sixth degree polynomial equation in general forms basing on new theorems, where the possibility to calculate the six roots of any sixth degree equation nearly simultaneously. The proposed roots for…
In this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second--order linear partial differential equations, which are admissible potentially…
Model theoretic ranks of solutions to Painleve equations are calculated, and the type of the generic solution of the second Painleve equation is shown to be disintegrated, strengthening a theorem of Nagloo. A question of Hrushovski and…
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…
We extend Painlev\'e's determinateness theorem to the case of first order ordinary differential equations in the complex domain with known terms allowed be multivalued in the dependent variable as well; multivaluedness is supposed to be…
We present a determinant expression for a family of classical transcendental solutions of the Painlev\'e V and the Painlev\'e VI equation. Degeneration of these solutions along the process of coalescence for the Painlev\'e equations is…
In this paper we study different Hamiltonian systems with polynomial and rational Hamiltonians associated with the generic third Painlev\'e equation and present explicit birational transformations relating them.
We develop a systematic approach to deriving rational solutions and obtaining classification of their parameters for dressing chains of even N periodicity or equivalently $A^{(1)}_{N-1}$ invariant Painlev\'e equations. This construction…
We offer elementary proofs for fundamental properties of solutions to the homogeneous second Painlev\'e equation.
We give an explicit determinant formula for a class of rational solutions of the Painlev\'e V equation in terms of the universal characters.
Our aim in this paper is to prove, under some growth conditions on the datas, the solvability in a Gevrey class of a polynomially nonlinear functional differential equation.