The Role of Salvatore Pincherle in the Development of Fractional Calculus

We revisit two contributions by Salvatore Pincherle (Professor of Mathematics at the University of Bologna from 1880 to 1928) published (in Italian) in 1888 and 1902 in order to point out his possible role in the development of Fractional Calculus. Fractional Calculus is that branch of mathematical analysis dealing with pseudo-differential operators interpreted as integrals and derivatives of non-integer order. Even if the former contribution (published in two notes on Accademia dei Lincei) on generalized hypergeomtric functions does not concern Fractional Calculus it contains the first example in the literature of the use of the so called Mellin–Barnes integrals. These integrals will be proved to be a fundamental task to deal with all higher transcendental functions including the Meijer and Fox functions introduced much later. In particular, the solutions of differential equations of fractional order are suited to be expressed in terms of these integrals. In the second paper (published on Accademia delle Scienze di Bologna), the author is interested to insert in the framework of his operational theory the notion of derivative of non integer order that appeared in those times not yet well established. Unfortunately, Pincherle’s foundation of Fractional Calculus seems still ignored.