8 problems

Discrete Painlevé integrability conjecture. All the difference equations arising in this way are discrete Painlevé equations satisfying integrability criteria such as singularity c…

Let $\kappa,\tau\in\mathbb{R}$, and consider the dP$_{\rm I}^{(2)}$ equation for a sequence $\{\beta_n(\kappa,\tau)\}_{n\geq 0}$, with initial values specified by the source. A sol…

Let $m>3$, and let $\beta_n$ denote the recurrence coefficient associated with the Freud unitary ensemble. After a minor change of variables, possibly $w_n=4\beta_n/a^2$, the recur…

The second canonical form is an additive fourth-order difference equation with a continuum limit given by the fourth-order Painlevé I equation, and its discrete Lagrangian and inva…

Let $b_n(t)$ and $a_n(t)$ denote the positive solutions and associated quantities introduced in the paper, with $n$ a nonnegative integer. For fixed $t>0$, the expressions…

Linear recurrence conjecture. The iterates satisfy a constant-coefficient linear recurrence relation of order $4(N-1)(N-2)$. For odd $N$, it has the form

The authors study discrete Painlevé II equations over finite fields by extending the domain through spaces of initial conditions and by reduction modulo a prime from a local field.…

The discrete Painlevé II equation is studied over finite fields by reducing modulo a prime number from a $p$-adic number field, using almost good reduction, an arithmetic analogue…