In this paper, we introduce a new class of harmonic univalent functions with respect to k-symmetric points by using a newly-defined q-analog of the derivative operator for complex harmonic functions. For this harmonic univalent function class, we derive a sufficient condition, a representation theorem, and a distortion theorem. We also apply a generalized q-Bernardi–Libera–Livingston integral operator to examine the closure properties and coefficient bounds. Furthermore, we highlight some known consequences of our main results. In the concluding part of the article, we have finally reiterated the well-demonstrated fact that the results presented in this article can easily be rewritten as the so-called $(p,q)$-variations by making some straightforward simplifications, and it will be an inconsequential exercise, simply because the additional parameter p is obviously unnecessary.

中文翻译：

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一类涉及某个q-导数算子的k-对称调和函数

在本文中，我们通过使用复调和函数的导数算子的新定义q类比，引入了一类关于k对称点的新调和单价函数。对于这个调和单价函数类，我们推导出一个充分条件、一个表示定理和一个畸变定理。我们还应用了广义q -Bernardi-Libera-Livingston 积分算子来检查闭包属性和系数界限。此外，我们强调了我们主要结果的一些已知后果。在文章的结尾部分，我们终于重申了一个充分证明的事实，即本文中呈现的结果可以很容易地改写为所谓的$(磷,q)$-variations 通过进行一些直接的简化，这将是一个无关紧要的练习，仅仅因为附加参数p显然是不必要的。