In this paper, we introduce the concept of n-dimensional Diamond-Alpha integral on time scales. In particular, it transforms into multiple Delta, Nabla and mixed integrals by taking different values of alpha. Some of its properties are explored, and the relationship between it and the multiple mixed integral is provided. As an application, we establish some weighted Ostrowski type inequalities through the new integral. These new inequalities expand some known inequalities in the monographs and papers, and in addition, furnish some other interesting inequalities. Examples of Ostrowski type inequalities are posed in detail at the end of the paper.

中文翻译：

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一般形式的多个Diamond-α积分及其性质，应用

在本文中，我们介绍了时标上的n维Diamond-Alpha积分的概念。特别是，通过采用不同的alpha值，它可以转换为多个Delta，Nabla和混合积分。探索了它的一些性质，并提供了它与多重混合积分之间的关​​系。作为应用程序，我们通过新的积分建立了一些加权的Ostrowski型不等式。这些新的不等式扩大了专着和论文中一些已知的不等式，此外，还提供了其他一些有趣的不等式。Ostrowski型不等式的示例在本文末尾详细列出。