Science and Education Publishing
American Journal of Applied Mathematics and Statistics
2328-7292
3
3
2015
06
09
A Unified Formula for The nth Derivative and The nth Anti-Derivative of the Bessel Function of Real Orders
100
104
EN
Mhenni M.
Benghorbal
Montreal, QC, Canada
AJAMS2015333
10.12691/ajams-3-3-3
2015
04
20
2015
05
21
2015
06
09
A complete solution to the problem of finding the nth derivative and the nth anti-derivative of elementary and special functions has been given. It deals with the problem of finding formulas for the nth derivative and the nth anti-derivative of elementary and special functions. We do not limit n to be an integer, it can be a real number. In general, the solution is given through unified formulas in terms of the Fox H-function and the Miejer G-function which, in many cases, can be simplified to less general functions. This, in turn, makes the first real use of these two special functions in the literature and shows the need of such functions. In this talk, we would like to present the idea on the Bessel function which is a well known special function. One of the key points in this work is that the approach does not depend on integration techniques. We adopt the classical definitions for generalization of differentiation and integration. Namely, the nth order of differentiation is found according to the Riemann-Liouville definition where (k ? 1 < n < k) and . The generalized Cauchy n-fold integral is adopted for the nth order of integration