# On the Extension of the Bessel Functions of the First Kind with Applications to some Mathematical Integrals and Transforms

On the Extension of the Bessel Functions of the First Kind with Applications to some Mathematical Integrals and Transforms. PhD thesis, King Fahd University of Petroleum and Minerals.

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PHD_Thesis_for_AbdulKhaleg_AlBaiyat.pdf - Accepted Version

Bessel Functions that have been known in the literature and are applied in many areas in engineering fields will be extended here. They will be defined through the extension of the recently defined extended confluent hypergeometric functions. We will provide some insights into the original function which is directly related to the Bessel function and the modified Bessel function of the first kind. As such, its extension provides an extension of the two Bessel functions of the first kind as well. In particular, it provides an integral representation which {\em yields an integral formula relating the standard Bessel function of any order to the Bessel function of order 1}. Indeed, as the extension is carried out through the extension of the beta function, some important results have been derived for this function too. In particular, it is shown that the difference between the function with first variable shifted by any integer $n\geq1$ and that of the function with the first variable shifted by one is the same as the corresponding difference for the second variable.\\ As the application part is the most important part of any extension, we have applied our extension to develop the two known integral identities (Lipschitz and Hankel's) into closed mathematical forms. The Laplace transform has also been obtained for the extended Bessel function of the first kind of order zero. {\em Interestingly, the closed form of the transform is expressed in terms of the extended hypergeometric function, which has also been recently introduced}. Studies of the extended Bessel function of the first kind as the asymptotic behavior and the Mellin-Barnes integral have also been covered in this work. On the continuation of the application part, we prove that the extended Bessel function of the first kind satisfies a non homogeneous second order differential equation. The classical Bessel differential equation satisfied by the standard Bessel function is deduced as a special case of this non homogeneous differential equation.