The goal of this paper is to demonstrate many explicit computational formulas and relations involving the Changhee polynomials and numbers and their differential equations with the help of functional equations and partial derivative equations for generating functions of these polynomials and numbers. These formulas also include the Euler polynomials, the Stirling numbers, the Bernoulli numbers and polynomials of the second kind, the Changhee polynomials of higher order, and the Daehee polynomials of higher order, which are among the well known polynomial families. By using PDEs of these generating functions, not only some recurrence relations for derivative formulas of the Changhee polynomials of higher order, but also two open problems for partial derivative equations for generating functions are given. Moreover, by using functional equations of the generating functions, two inequalities including combinatorial sums, the Changhee numbers of negative order, and the Stirling numbers of the second kind are provided. Finally, further remarks and observations for the results of this paper are given.

中文翻译：

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Changhee多项式的计算公式及其功能和微分方程的推导

本文的目的是借助功能方程和偏导数方程来证明许多涉及Changhee多项式和数字及其微分方程的显式计算公式和关系，以生成这些多项式和数字的函数。这些公式还包括众所周知的多项式族，其中包括欧拉多项式，斯特林数，第二类伯努利数和多项式，高阶的Changhee多项式和高阶的Daeeee多项式。利用这些生成函数的PDE，不仅给出了高阶Changhee多项式的导数公式的一些递推关系，而且还给出了生成函数的偏导数方程的两个开放问题。此外，通过使用生成函数的函数方程，提供了两个不等式，包括组合和，负阶的Changhee数和第二类的Stirling数。最后，对本文的结果作了进一步的评论和观察。