In this paper, our leading objective is to relate the fractional integral operator known as -transform with the -extended Mathieu series. We show that the -transform turns to the classical Laplace transform; then, we get the integral relating the Laplace transform stated in corollaries. As corollaries and consequences, many interesting outcomes are exposed to follow from our main results. Also, in this paper, we have converted the -transform into a classical Laplace transform by changing the variable ; then, we get the integral involving the Laplace transform.

中文翻译：

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使用拉普拉斯变换与扩展 Mathieu 级数相关的分数运算符

在本文中，我们的主要目标是将称为-变换的分数积分算子与-扩展的 Mathieu 级数联系起来。我们证明-变换变成了经典的拉普拉斯变换；然后，我们得到推论中陈述的与拉普拉斯变换相关的积分。作为推论和后果，从我们的主要结果中可以得出许多有趣的结果。此外，在本文中，我们通过改变变量将-变换转换为经典的拉普拉斯变换; 然后，我们得到涉及拉普拉斯变换的积分。