标题： 平面中的奇异行走和伯努利数 显示英文标题

标题： Singular walks in the quarter plane and Bernoulli numbers

摘要： 我们考虑在第一象限中的奇异（也称为 genus$0$）步进，并它们相关的生成函数$Q(x,y,t)$，这些生成函数枚举从原点开始的步进，固定终点（由空间变量$x$和$y$编码）和固定长度（由时间变量$t$编码）。 我们首先证明，前一个级数可以扩展到一个通用值$t$（在所有奇异模型中都成立），即$t=\frac{1}{2}$，并且我们提供了$Q(x,y,\frac{1}{2})$的概率解释。 作为第二步，我们改进了文献中的早期结果并证明，对于任何$t\in(0,\frac{1}{2}]$，$Q(x,y,t)$确实是微分超越的。 此外，我们证明$Q(x,y,\frac{1}{2})$是强微分超越的。 作为最后一步，我们证明对于某些模型，$Q(x,y,\frac{1}{2})$的级数展开直接与伯努利数相关。 这提供了其强微分超越性的第二个证明。 显示英文摘要

摘要： We consider singular (aka genus $0$) walks in the quarter plane and their associated generating functions $Q(x,y,t)$, which enumerate the walks starting from the origin, of fixed endpoint (encoded by the spatial variables $x$ and $y$) and of fixed length (encoded by the time variable $t$). We first prove that the previous series can be extended up to a universal value of $t$ (in the sense that this holds for all singular models), namely $t=\frac{1}{2}$, and we provide a probabilistic interpretation of $Q(x,y,\frac{1}{2})$. As a second step, we refine earlier results in the literature and show that $Q(x,y,t)$ is indeed differentially transcendental for any $t\in(0,\frac{1}{2}]$. Moreover, we prove that $Q(x,y,\frac{1}{2})$ is strongly differentially transcendental. As a last step, we show that for certain models the series expansion of $Q(x,y,\frac{1}{2})$ is directly related to Bernoulli numbers. This provides a second proof of its strong differential transcendence.