In this paper, by the Faà di Bruno formula and properties of Bell polynomials of the second kind, the authors reconsider the generating functions of Hermite polynomials and their squares, find an explicit formula for higher-order derivatives of the generating function of Hermite polynomials, and derive explicit formulas and recurrence relations for Hermite polynomials and their squares.

中文翻译：

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Hermite多项式的某些性质

本文通过第二类Bell多项式的Faàdi Bruno公式和性质，重新考虑了Hermite多项式及其平方的生成函数，找到了Hermite多项式生成函数的高阶导数的显式公式，并推导Hermite多项式及其平方的明确公式和递归关系。