Generalized fractional operators do not preserve periodicity

This work allows proving that the action of fractional derivatives and fractional integrals on periodic functions does not preserve the periodicity of any period. This result is proved not only for one type of fractional operator but also for the wide class of generalized fractional operators based on the Sonine condition, a class that encompasses the majority of the fractional operators commonly used. Moreover, for several specific fractional operators, we provide explicit representations of the derivatives and integrals of the sine function, showing that they are composed of a local periodic term and a non-local term, which is the cause of the loss of periodicity.