Fractional Operators Associated with the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi>ք</a:mi> </a:math>-Extended Mathieu Series by Using Laplace Transform

In this paper, our leading objective is to relate the fractional integral operator known as <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M2"> <a:msub> <a:mrow> <a:mi>P</a:mi> </a:mrow> <a:mrow> <a:mi>δ</a:mi> </a:mrow> </a:msub> </a:math> -transform with the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M3"> <c:mi>ք</c:mi> </c:math> -extended Mathieu series. We show that the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M4"> <e:msub> <e:mrow> <e:mi>P</e:mi> </e:mrow> <e:mrow> <e:mi>δ</e:mi> </e:mrow> </e:msub> </e:math> -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 <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" id="M5"> <g:msub> <g:mrow> <g:mi>P</g:mi> </g:mrow> <g:mrow> <g:mi>δ</g:mi> </g:mrow> </g:msub> </g:math> -transform into a classical Laplace transform by changing the variable <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" id="M6"> <i:mfenced open="(" close=")"> <i:mrow> <i:mfenced open="(" close=")"> <i:mrow> <i:mi mathvariant="normal">ln</i:mi> <i:mfenced open="[" close="]"> <i:mrow> <i:mfenced open="(" close=")"> <i:mrow> <i:mi>δ</i:mi> <i:mo>−</i:mo> <i:mn>1</i:mn> </i:mrow> </i:mfenced> <i:mi>s</i:mi> <i:mo>+</i:mo> <i:mn>1</i:mn> </i:mrow> </i:mfenced> </i:mrow> </i:mfenced> <i:mo>/</i:mo> <i:mfenced open="(" close=")"> <i:mrow> <i:mi>δ</i:mi> <i:mo>−</i:mo> <i:mn>1</i:mn> </i:mrow> </i:mfenced> </i:mrow> </i:mfenced> <i:mo>⟶</i:mo> <i:mi>s</i:mi> </i:math> ; then, we get the integral involving the Laplace transform.