On the importance of conjugation relations in fractional calculus

Abstract

It is useful to understand how the various operators of fractional calculus relate to each other, especially relations between newly defined operators and classical well-studied ones. This paper focuses on an important type of such relationship, namely conjugation relations, also called transmutation relations. We define a general abstract setting in which such relations are relevant, and indicate how they can be used to prove many results easily in general settings such as fractional calculus with respect to functions and weighted fractional calculus.