A catalogue of semigroup properties for integral operators with Fox–Wright kernel functions

Abstract

AbstractThe Fox–Wright function is a very general form of function, covering many families of special functions as particular cases. Any special function can be used as a kernel for a fractional integral operator, but which of these operators will satisfy desiderata such as a semigroup property for composition? This paper provides a rigorous categorisation of all such fractional integral operators which have a semigroup property in any of their parameters. We discover that nearly all possible semigroup properties arise from the Chu–Vandermonde identity, with the Prabhakar fractional calculus emerging as one special case. For any integral operator with such a semigroup property, it is possible to construct a complete model of fractional calculus, including both integral and derivative operators which interact with each other in a natural way.