Bessel-Clifford operators and the generalized fractional calculus

The theory of hyper-Bessel differential operators of arbitrary order m > 1, has been shown to be closely related to the Meijer's G - functions. These generalized hypergeometric functions incorporate as particular cases the basic elementary functions and almost all the special functions of mathematical physics. Such functions are the kernels of the integral operators and transforms as well as the solutions of the Bessel-Clifford differential equations of arbitrary order. However, most of the operational calculi, integral transforms and solutions to the Bessel type differential equations developed by different authors concern special cases mainly of order m = 2 when the role of these special functions is not evident. Here, we give an example of a third order Bessel type operator and emphasize on the use of the generalized fractional calculus and G - functions. Main attention is paid to the corresponding Laplace-Obrechkoff type integral transform with some examples of its applications for solving initial value problems for Bessel-Clifford differential equations of third order. Many initial and boundary value problems of mathematical physics are related with these type of operators. These generalized operators were introduced by Dimovski, who developed operational calculi, integral transforms and transmutation operators for them. Later, these investigations were extended principally by Kiryakova and many other authors.