论文标题
钟数和亲戚的差异超越：GALOIS理论方法
Differential transcendence of Bell numbers and relatives: a Galois theoretic approach
论文作者
论文摘要
在2003年，克拉扎尔（Klazar）证明了贝尔数字序列的普通生成函数在字段上是差异性的，$ \ mathbb {c}（\ {t \}）$以$ 0 $ $ 0 $。我们表明，克拉扎尔的结果是一种普遍现象的实例，可以使用差异galois理论以紧凑的方式证明。我们介绍了该理论的主要原理，以证明$ \ Mathbb {c}（\ {t \}）$的差异超越性的一般结果，我们将我们应用于生成功能的许多其他（无限类）示例，包括Klazar认为非常特殊的情况。我们的大多数例子都属于谢弗（Sheffer）的班级，在翁布拉尔（Umbral）演算中进行了良好的研究。他们都为pak-yeliussizov猜想提供了具体的证据，根据该序列，其普通和指数产生函数的序列都以多项式系数满足了非线性差分方程，必须满足具有多项式系数的线性复发。
In 2003 Klazar proved that the ordinary generating function of the sequence of Bell numbers is differentially transcendental over the field $\mathbb{C}(\{t\})$ of meromorphic functions at $0$. We show that Klazar's result is an instance of a general phenomenon that can be proven in a compact way using difference Galois theory. We present the main principles of this theory in order to prove a general result about differential transcendence over $\mathbb{C}(\{t\})$, that we apply to many other (infinite classes of) examples of generating functions, including as very special cases the ones considered by Klazar. Most of our examples belong to Sheffer's class, well studied notably in umbral calculus. They all bring concrete evidence in support to the Pak-Yeliussizov conjecture, according to which a sequence whose both ordinary and exponential generating functions satisfy nonlinear differential equations with polynomial coefficients necessarily satisfies a linear recurrence with polynomial coefficients.
