Carlitz initiated a study of degenerate Bernoulli and Euler numbers and polynomials which is the pioneering work on degenerate versions of special numbers and polynomials. In recent years, studying degenerate versions regained lively interest of some mathematicians. The purpose of this paper is to study degenerate Bell polynomials by using umbral calculus and generating functions. We derive several properties of the degenerate Bell polynomials including recurrence relations, Dobinski-type formula, and derivatives. In addition, we represent various known families of polynomials such as Euler polynomials, modified degenerate poly-Bernoulli polynomials, degenerate Bernoulli polynomials of the second kind, and falling factorials in terms of degenerate Bell polynomials and vice versa.

中文翻译：

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与本影演算相关的简并贝尔多项式

Carlitz发起了对退化伯努利和欧拉数和多项式的研究，这是特殊数字和多项式的退化形式的开创性工作。近年来，研究退化版本重新引起了一些数学家的兴趣。本文的目的是通过使用本影演算和生成函数来研究退化的贝尔多项式。我们推导了简并的贝尔多项式的几个属性，包括递归关系，Dobinski型公式和导数。此外，我们代表了各种已知的多项式族，例如Euler多项式，修改的简并的Poly-Bernoulli多项式，第二类的简并的Bernoulli多项式以及就简并的Bell多项式而言的降阶因式，反之亦然。