In this paper, we introduce the r-central factorial numbers with even indices of the first and second kind as extended versions of the central factorial numbers with even indices of both kinds. We obtain several fundamental properties and identities related to these numbers. The connections between the new numbers and the Stirling numbers are presented. In addition, we give the probability distribution of the unsigned r-central factorial numbers with even indices. Finally, we consider the r-central factorial matrices and study some of their properties.

中文翻译：

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带有偶数索引的r中心阶乘数

在本文中，我们介绍具有第一种和第二种偶数索引的r中心阶乘数，作为具有两种偶数索引的中心阶数的扩展版本。我们获得了一些与这些数字有关的基本特性和身份。介绍了新数字和斯特林数字之间的联系。另外，我们给出具有偶数索引的无符号r-中心阶乘数的概率分布。最后，我们考虑r-中心阶乘矩阵，并研究它们的一些特性。