Recently, the nth Lah–Bell number was defined as the number of ways a set of n elements can be partitioned into nonempty linearly ordered subsets for any nonnegative integer n. Further, as natural extensions of the Lah–Bell numbers, Lah–Bell polynomials are defined. We study Lah–Bell polynomials with and without the help of umbral calculus. Notably, we use three different formulas in order to express various known families of polynomials such as higher-order Bernoulli polynomials and poly-Bernoulli polynomials in terms of the Lah–Bell polynomials. In addition, we obtain several properties of Lah–Bell polynomials.

中文翻译：

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Lah-Bell多项式的某些恒等式

最近，第n个Lah-Bell数定义为将n个元素的集合划分为任意非负整数n的非空线性有序子集的方式的数量。此外，作为Lah-Bell数的自然扩展，定义了Lah-Bell多项式。我们研究带或不带本影演算的Lah-Bell多项式。值得注意的是，我们使用三个不同的公式来表示各种已知的多项式族，例如根据Lah-Bell多项式来表示高阶Bernoulli多项式和poly-Bernoulli多项式。此外，我们获得了Lah-Bell多项式的几个属性。