The hunting for the discrete Painlev\'e VI is over

In the previous work we introduced the higher order $q$-Painlev\'{e} system $q$-$P_{(n+1,n+1)}$ as a generalization of the Jimbo-Sakai's $q$-Painlev\'{e} VI equation. It is derived from a $q$-analogue of the Drinfeld-Sokolov hierarchy of type $A^{(1)}_{2n+1}$ and admits a particular solution in terms of the Heine's $q$-hypergeometric function ${}_{n+1}\phi_n$. However the obtained system is insufficient as a generalization of $q$-$P_{\rm{VI}}$ due to some reasons. In this art...

For transcendental functions that solve non-linear $q$-difference equations, the best descriptions available are the ones obtained by expansion near critical points at the origin and infinity. We describe such solutions of a $q$-discrete Painlev\'e equation, with 7 parameters whose initial value space is a rational surface of type $A_1^{(1)}$. The resultant expansions are shown to approach series expansions of the classical sixth Painlev\'e equation in the continuum limit.

Rational solutions for a $q$-difference analogue of the Painlev\'e III equation are considered. A Determinant formula of Jacobi-Trudi type for the solutions is constructed.

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 viewpoint of affine Weyl group symmetry. In particular, the mechanism of apparent inconsistency between the hypergeometric solutions to both equations is clarified by using factorization of difference operators and the $\tau$ functions.

An ultradiscrete analog of the Bessel function is constructed by taking the ultradiscrete limit for a $q$-difference analog of the Bessel function. Then, a direct relationship between a class of special solutions for the ultradiscrete Painlev\'{e} III equation and those of the discrete Painlev\'{e} III equation which have a determinantal structure is established.

In the current paper we study the $q$-analogue introduced by Jimbo and Sakai of the well known Painlev\'e VI differential equation. We explain how it can be deduced from a $q$-analogue of Schlesinger equations and show that for a convenient change of variables and auxiliary parameters, it admits a $q$-analogue of Hamiltonian formulation. This allows us to show that Sakai's $q$-analogue of Okamoto space of initial conditions for $qP_\mathrm{VI}$ admits the differential Okamoto...

A q-difference analogue of the fourth Painlev\'e equation is proposed. Its symmetry structure and some particular solutions are investigated.

A $q$-difference analog of the sixth Painlev\'e equation is presented. It arises as the condition for preserving the connection matrix of linear $q$-difference equations, in close analogy with the monodromy preserving deformation of linear differential equations. The continuous limit and special solutions in terms of $q$-hypergeometric functions are also discussed.

A class of special solutions are constructed in an intuitive way for the ultradiscrete analog of $q$-Painlev\'e II ($q$-PII) equation. The solutions are classified into four groups depending on the function-type and the system parameter.

A Riemann-Hilbert problem for a $q$-difference Painlev\'e equation, known as $q\textrm{P}_{\textrm{IV}}$, is shown to be solvable. This yields a bijective correspondence between the transcendental solutions of $q\textrm{P}_{\textrm{IV}}$ and corresponding data on an associated $q$-monodromy surface. We also construct the moduli space of $q\textrm{P}_{\textrm{IV}}$ explicitly.