The aim of this paper is to construct generating functions for a new family of polynomials, which are called parametric Hermite‐based Milne–Thomson type polynomials. Many properties of these polynomials with their generating functions are investigated. These generating functions give us generalization of some well‐known generating functions for special polynomials such as Hermite type polynomials, Milne–Thomson type polynomials, and Apostol type polynomials. Using the Euler formula, functional equation method for generating function, and differential operator technique, we give relations among parametric Hermite‐based Milne–Thomson type polynomials, the Bernoulli numbers, the Euler numbers, the Chebyshev polynomials, the Bernstein basis functions, homogeneous harmonic polynomials, and parametric kinds of Apostol type polynomials. Moreover, some computational formulas for these polynomials are derived. Finally, using Wolfram Mathematica version 12.0, some of these polynomials and their generating functions are illustrated by their plots under the special conditions. Potential relationships and connections of this paper's results with the results of previous and future research are pointed out.

中文翻译：

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新类基于Hermite的Milne-Thomson型多项式的计算公式和恒等式：使用Euler公式对生成函数进行分析

本文的目的是构造一个新的多项式族的生成函数，这些族称为基于参数Hermite的Milne-Thomson型多项式。研究了这些多项式的许多性质及其生成函数。这些生成函数使我们对特殊多项式的一些著名生成函数进行了概括，例如Hermite型多项式，Milne-Thomson型多项式和Apostol型多项式。使用Euler公式，用于生成函数的函数方程方法和微分算子技术，我们给出了基于参数Hermite的Milne-Thomson型多项式，Bernoulli数，Euler数，Chebyshev多项式，Bernstein基函数，齐次谐波之间的关系多项式和Apostol型多项式的参数种类。而且，推导了这些多项式的一些计算公式。最后，使用Wolfram Mathematica 12.0版，通过在特殊条件下的图来说明其中的一些多项式及其生成函数。指出了本文结果与先前和未来研究结果之间的潜在关系和联系。