ABSTRACT In this article, we first give some basic properties of generalized Hermite polynomials associated with parabolic cylinder functions. We next use Weisnerś group theoretic method and operational rules method to establish new generating functions for these generalized Hermite polynomials. The operational methods we use allow us to obtain unilateral, bilinear and bilateral generating functions by using the same procedure. Applications of generating functions obtained by Weisnerś group theoretic method are discussed.

中文翻译：

---

生成与抛物柱面函数相关的广义 Hermite 多项式的函数

摘要 在本文中，我们首先给出与抛物柱面函数相关的广义 Hermite 多项式的一些基本性质。我们接下来使用 Weisnerś 群论方法和运算规则方法为这些广义 Hermite 多项式建立新的生成函数。我们使用的操作方法允许我们使用相同的程序获得单边、双线性和双边生成函数。讨论了由 Weisnerś 群论方法获得的生成函数的应用。