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We consider matrix orthogonal polynomials related to Jacobi type matrices of weights that can be defined in terms of a given matrix Pearson equation. Stating a Riemann-Hilbert problem we can derive first and second order differential…
We discuss a family of Hamiltonians given by particular rational extensions of the singular oscillator in two-dimensions. The wave functions of these Hamiltonians can be expressed in terms of products of Laguerre and exceptional Jacobi…
Starting from the notion of discriminantly separable polynomials of degree two in each of three variables, we construct a class of integrable dynamical systems. These systems can be integrated explicitly in genus two theta-functions in a…
We give an explicit determinant formula for a class of rational solutions of a q-analogue of the Painlev\'e V equation. The entries of the determinant are given by the continuous q-Laguerre polynomials.
Several recently discovered properties of multiple families of special polynomials (some orthogonal and some not) that satisfy certain differential, difference or q-difference equations are reviewed. A general method of construction of…
Recursive algebraic construction of two infinite families of polynomials in $n$ variables is proposed as a uniform method applicable to every semisimple Lie group of rank $n$. Its result recognizes Chebyshev polynomials of the first and…
We study a sequence of polynomials orthogonal with respect to a one parameter family of weights $$ w(x):=w(x,t)=\rex^{-t/x}\:x^{\al}(1-x)^{\bt},\quad t\geq 0, $$ defined for $x\in[0,1].$ If $t=0,$ this reduces to a shifted Jacobi weight.…
We present a new approach to determine the rational solutions of the higher order Painleve equations associated to periodic dressing chain systems. We obtain new sets of solutions, giving determinantal representations indexed by specific…
New methods for derivation of Bell polynomials of the second kind are presented. The methods are based on an ordinary generating function and its composita. The relation between a composita and a Bell polynomial is demonstrated. Main…
Starting from the second Painlev\'{e} equation, we obtain Painlev\'{e} type equations of higher order by using the singular point analysis.
The aim of this paper is to give two new algorithms, which are elimination free, to find polynomial and rational solutions for a given holonomic system associated to a set of linear differential operators in the Weyl algebra D = k<x_1, ...,…
This paper considers functional series whose terms are higher-order derivatives of Chebyshev polynomials of the second kind, where the degree of the polynomial is related to the order of the derivative. Analytic summation is used to…
The Painlev\'{e} equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of ``classical" weights multiplied by suitable ``deformation factors", usually dependent on a…
Bilinear systems of equations are defined, motivated and analyzed for solvability. Elementary structure is mentioned and it is shown that all solutions may be obtained as rank one completions of a linear matrix polynomial derived from…
Our purpose in this paper is to study when a planar differential system polynomial in one variable linearizes in the sense that it has an inverse integrating factor which can be constructed by means of the solutions of linear differential…
Regarding the resolution of singularities for the differential equations of Painlev\'e type, there are important differences between the second-order Painlev\'e equations and those of higher order. Unlike the second-order case, in higher…
We find and study a two-parameter family of coupled Painlev\'e II systems in dimension four with affine Weyl group symmetry of several types. Moreover, we find a three-parameter family of polynomial Hamiltonian systems in two variables…
We solve a special type of linear systems with coefficients in multivariate polynomial rings. These systems arise in the computation of parametric Bernstein-Sato polynomials associated with certain hypergeometric ideals in the Weyl algebra.
Using Chebyshev polynomialsof both kinds, we construct rational fractions which are convergents of the smallest root of $x^2-\alpha x+1$ for $\alpha=3,4,5,\dots$.Some of the underlying identities suggest an identity involving…
A relationship between Painleve systems and infinite-dimensional integrable hierarchies is studied. We derive a class of higher order Painleve systems from Drinfeld-Sokolov (DS) hierarchies of type A by similarity reductions. This result…