标题： 延迟Painlevé-I方程，相关多项式和Masur-Veech体积 显示英文标题

标题： Delay Painlevé-I equation, associated polynomials and Masur-Veech volumes

摘要： 我们研究了Shabat的dressing链通过延迟周期约化得到的第一个Painlevé方程的延迟微分模拟。 我们构建了该方程的形式整体解，并引入了一类新的多项式（称为Bernoulli-Catalan多项式），这些多项式由Catalan类型的非线性递推定义，并且与Bernoulli和Euler多项式具有相似的性质。 我们还讨论了亚纯解，并通过类型为$A_1^{(1)}$的仿射Weyl群来描述这个延迟Painlevé-I方程的奇点结构。 作为应用，我们展示了与计算亚纯二次微分模空间的Masur-Veech体积问题之间的联系，重新推导了一些已知公式。 显示英文摘要

摘要： We study a delay-differential analogue of the first Painlev\'e equation obtained as a delay periodic reduction of Shabat's dressing chain. We construct formal entire solutions to this equation and introduce a new family of polynomials (called Bernoulli-Catalan polynomials), which are defined by a nonlinear recurrence of Catalan type, and which share properties with Bernoulli and Euler polynomials. We also discuss meromorphic solutions and describe the singularity structure of this delay Painlev\'e-I equation in terms of an affine Weyl group of type $A_1^{(1)}$. As an application we demonstrate the link with the problem of calculation of the Masur-Veech volumes of the moduli spaces of meromorphic quadratic differentials by re-deriving some of the known formulas.