The aim of this paper is to apply trigonometric functions with functional equations of generating functions. Using the resulted new equations and formulas from this application, we obtain many special numbers and polynomials such as the Stirling numbers, Bernoulli and Euler type numbers, the array polynomials, the Catalan numbers, and the central factorial numbers. We then introduce combinatorial sums related to these special numbers and polynomials. Moreover, we gave some remarks that relates our new findings from this paper to the relations found in earlier studies.

中文翻译：

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特殊数字和多项式的新计算公式，源于将三角函数应用于生成函数

本文的目的是将三角函数与生成函数的函数方程一起应用。使用由此应用程序产生的新方程式和公式，我们获得了许多特殊的数和多项式，例如斯特林数，伯努利和欧拉类型数，数组多项式，加泰罗尼亚数和中央阶乘数。然后，我们介绍与这些特殊数字和多项式有关的组合和。此外，我们作了一些评论，将本文的新发现与早期研究中发现的关系联系起来。