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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 study self-adjoint matrix polynomial equations in a single variable and prove existence of self-adjoint solutions under some assumptions on the leading form. Our main result is that any self-adjoint matrix polynomial equation of odd…
This expository article written for the Notices of the American Mathematical Society provides an overview of transcendental functions arising as solutions of the discrete Painlev\'e equations, for which the developments of the last two…
The solutions of the discrete Painlev\'e equation I were constructed in terms of elliptic and hyperelliptic $\psi$ functions for algebraic curves of genera one and two. For the case of genus two, there appear higher order difference…
By making use of a recently developed method to solve linear differential equations of arbitrary order, we find a wide class of polynomial solutions to the Heun equation. We construct the series solution to the Heun equation before…
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…
We show that symmetric polynomials previously introduced by the author satisfy a certain differential equation. After a change of variables, it can be written as a non-stationary Schr\"odinger equation with elliptic potential, which is…
In this paper we show that an arbitrary solution of one ordinary difference equation is also a solution for a hierarchy of integrable difference equations. We also provide an example of such a solution that is related to sequence generated…
For large classes of systems of polynomial nonlinear PDEs necessary and sufficient conditions are given for the existence of solutions which are discontinuous across hyper-surfaces. These PDEs contain the Navier-Stokes equations, as well as…
In this paper, we analyze the asymptotic behaviour of the poles of certain rational solutions of the fifth Painlev\'e equation. These solutions are constructed by relating the corresponding tau function to a Hankel determinant of a certain…
In the work we use integral formulas for calculating the monodromy data for the Painlev\'e-2 equation. The perturbation theory for the auxiliary linear system is constructed and formulas for the variation of the monodromy data are obtained.…
The critical and asymptotic behaviors of solutions of the sixth Painlev\'e equation, an their parametrization in terms of monodromy data, are synthetically reviewed. The explicit formulas are given. This paper has been withdrawn by the…
In this paper we obtain explicit expressions for tau-functions related to Picard type solutions of the Painlev\'e VI equation in terms of theta functions and their derivatives.
A general solution for a second-order linear constant coefficient dynamic equation with polynomial forcing on time scales is given.
We present various constructions of sequences of polynomials satisfying the Binomial Theorem in finite characteristic based on the theory of additive polynomials. Various actions on these constructions are also presented. It is an open…
In this paper, we study linear differential equations arising from Bessel polynomials and their applications. From these linear differential equations, we give some new and explicit identities for Bessel polynomials.
This article deals with the second order linear differential equations with entire coefficients. We prove some results involving conditions on coefficients so that the order of growth of every non-trivial solution is infinite.
We show that the discrete Painlev\'e-type equations arising from quantum minimal surfaces are equations for recurrence coefficients of orthogonal polynomials for indefinite hermitian products. As a consequence, we obtain an explicit formula…
A Lax pair for the additive difference Painlev\'e equation of type $E_7^{(1)}$ is explicitly obtained as certain linear difference equations of scalar form. The compatibility of the Lax pair is proved by using certain characterization of…
Algebraic solutions of the sixth Painleve equation can be computed using pullback transformations of hypergeometric equations with respect to specially ramified rational coverings. In particular, as was noticed by the second author and…