Related papers: A study on the fourth q-Painlev\'e equation
We propose an elemantary approach to Zudilin's q-question about Schmidt's problem [Electron. J. Combin. 11 (2004), #R22], which has been solved in a previous paper [Acta Arith. 127 (2007), 17--31]. The new approach is based on a q-analogue…
The discrete Painlev\'e equations have mathematical properties closely related to those of the differential Painlev\'e equations. We investigate the appearance of elliptic functions as limiting behaviours of $q$-Painlev\'e transcendents,…
A difference q-analogue of the dressing chain is discussed in this paper.
We consider the q-Painlev\'e III equation arising from the birational representation of the affine Weyl group of type $(A_2 + A_1)^{(1)}$. We study the reduction of the q-Painlev\'e III equation to the q-Painlev\'e II equation from the…
Completely solved the equivalence problem for the "`Painleve 34"' equation.
We use the middle convolution to obtain some old and new algebraic solutions of the Painlev\'e VI equations.
For the fourth Painlev\'e transcendents we derive elliptic asymptotic representations, which were announced by late Professor Kapaev without proofs. Then we newly obtain related results including the correction function.
We study some q-analogues of the Racah polynomials and some of their applications in the theory of representation of quantum algebras.
In this paper, we study some properties of the q-Appell polynomials, including the recurrence relations and the q-difference equations which extend some known calssical (q=1) results. We also provide the recurrence relations and the…
B\"acklund transformations (BTs) for ordinary differential equations (ODEs), and in particular for hierarchies of ODEs, are a topic of great current interest. Here we give an improved method of constructing BTs for hierarchies of ODEs. This…
We study a class of fourth order geometric equations defined on a 4-dimensional compact Riemannian manifold which includes the Q-curvature equation. We obtain sharp estimates on the difference near the blow-up points between a bubbling…
Following an idea due to J. Bernoulli, we explore the q-analogue of the sums of powers of consecutive integers.
This short review is an introduction to a great variety of methods, the collection of which is called the Painlev\'e analysis, intended at producing all kinds of exact (as opposed to perturbative) results on nonlinear equations, whether…
We introduce a $q$-analog of the polyanalytic Bargmann transform on $\mathbb{C}$.
Symmetries and solutions of the Painleve IV equation are presented in an alternative framework which provides the bridge between the Hamiltonian formalism and the symmetric Painleve IV equation. This approach originates from a method…
This paper is divided in two parts. In the first part we consider irregular singular analytic q-difference equations, with q\in ]0,1[, and we show how the Borel sum of a divergent solution of a differential equation can be uniformly…
We show that the topological recursion for the (semi-classical) spectral curve of the first Painlev\'e equation $P_{\rm I}$ gives a WKB solution for the isomonodromy problem for $P_{\rm I}$. In other words, the isomonodromy system is a…
Special polynomials associated with rational solutions of the second Painlev\'{e} equation and other members of its hierarchy are discussed. New approach, which allows one to construct each polynomial is presented. The structure of the…
Starting with a rational solution to Painleve' VI, coming from a Riccati equation, using Okamoto's theory a four-parametric rational solution is obtained.
Second order ordinary differential equations of the form $y'' = P(x,y) + 4 Q(x,y) y' + 6 R(x,y) y'^2 + 4 S(x,y) y'^3 + L(x,y) y'^4$ are considered and their point-expansions are constructed. Geometrical structures connected with these…