Fractional Operators Associated with the ք-Extended Mathieu Series by Using Laplace Transform

In this paper, our leading objective is to relate the fractional integral operator known as Pδ-transform with the ք-extended Mathieu series. We show that the Pδ-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 Pδ-transform into a classical Laplace transform by changing the variable lnδ−1s+1/δ−1⟶s; then, we get the integral involving the Laplace transform.