Fractional Bessel differential equation and fractional Bessel functions

Using a new simple and well-behaved definition of the fractional deriva-tive which is different from the Caputo and Riemann-Liouville fractional derivative and recently introduced by Khalil and others, we reformulate the second order Bessel dif-ferential equation in this new setting. In this article by the use of power series, one of the solution of the fractional differential equation is obtained. Moreover, we find the generating function and use it to prove some nice standard results and recurrence relations. Finally, we present some application and integral representations of Bessel functions of fractional type including sines and cosines.

Automated summary

In this article, a new definition of the fractional derivative is used to reformulate the second order Bessel differential equation. The solution of the fractional differential equation is obtained using power series. Additionally, the generating function is found and used to prove some standard results and recurrence relations. Finally, applications and integral representations of Bessel functions of fractional type, including sine and cosine, are presented.