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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…
Two types of determinant representations of the rational solutions for the Painlev\'e II equation are discussed by using the bilinear formalism. One of them is a representation by the Devisme polynomials, and another one is a Hankel…
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.
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
We analyze the polynomial solutions of the linear differential equation $p_2(x)y''+p_1(x)y'+p_0(x)y=0$ where $p_j(x)$ is a $j^{\rm th}$-degree polynomial. We discuss all the possible polynomial solutions and their dependence on the…
This work reports and classifies the most general construction of rational quantum potentials in terms of the generalized Hermite polynomials. This is achieved by exploiting the intrinsic relation between third-order shape-invariant…
This paper focuses on the construction of rational solutions for the $A_{2n}$ Painlev\'e system, also called the Noumi-Yamada system, which are considered the higher order generalizations of PIV. In this even case, we introduce a method to…
In this paper we follow the general approach, proposed earlier by the first author, which is derived from the invariant theory field and provides a way of obtaining of the polynomial identities for any arbitrary polynomial family. We…
We have been working in many aspects of the problem of analyzing, understanding and solving ordinary differential equations (first and second order). As we have extensively mentioned, while working in the Darboux type methods, the most…
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…
Recurrence coefficients of semi-classical orthogonal polynomials (orthogonal polynomials related to a weight function $w$ such that $w'/w$ is a rational function) are shown to be solutions of non linear differential equations with respect…
We present certain general structures related to the solutions of Painlev\'e equation II and to the solutions of the differential equation satisfied by the corresponding Hamiltonian equations, together with the tau functions. By taking…
We provide a complete classification and an explicit representation of rational solutions to the fourth Painlev\'e equation PIV and its higher order generalizations known as the $A_{2n}$-Painlev\'e or Noumi-Yamada systems. The construction…
We present a new approach to solving polynomial ordinary differential equations by transforming them to linear functional equations and then solving the linear functional equations. We will focus most of our attention upon the first-order…
In this manuscript we make major progress classifying algebraic relations between solutions of Painlev\'e equations. Our main contribution is to establish the algebraic independence of solutions of various pairs of equations in the…
We describe a family of polynomials discovered via a particular recursion relation, which have connections to Chebyshev polynomials of the first and the second kind, and the polynomial version of Pell's equation. Many of their properties…
We study Wronskians of Appell polynomials indexed by integer partitions. These families of polynomials appear in rational solutions of certain Painlev\'e equations and in the study of exceptional orthogonal polynomials. We determine their…
We review non-autonomous Hamiltonian systems, polynomial in two dependent variables, with the property that all of their solutions are meromorphic functions in the complex plane. These are related to known Hamiltonian systems with the…
Second-order polynomials generalize classical first-order ones in allowing for additional variables that range over functions rather than values. We are motivated by their applications in higher-order computational complexity theory,…
The new method for obtaining a variety of extensions of Hermite polynomials is given. As a first example a family of orthogonal polynomial systems which includes the generalized Hermite polynomials is considered. Apparently, either these…