Abstract

One of the most commonly discussed properties for any fractional differintegral operator is whether or not it has a semigroup property: for example, is the halfth integral of the halfth integral equal to the first integral? This question becomes more complicated in the setting of multi-parameter operators: for example, the Prabhakar integrals have a semigroup property in exactly two of their four parameters. We consider a general multi-parameter fractional integral operator with a Fox-Wright kernel function, and catalogue exhaustively all possible subsets of its parameters in which a semigroup property is possible. This integral operator is in general multi-dimensional, its dimension corresponding to the number of gamma functions on the denominator of the Fox-Wright function. In the cases where a semigroup property holds, we are able to construct a corresponding multi-dimensional fractional derivative operator which has the same natural inversion and analytic continuation properties as classical fractional derivatives.