The Role of Salvatore Pincherle in the Development of Fractional Calculus

Abstract

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.

Non-Gaussian diffusion of mixed origins

“…Diffusion processes with a stable distribution of particle jumps (space fractional diffusion) and power law waiting times between jumps (time fractional diffusion) have been investigated [51][52][53][54][55][56]. Here our approach allows us to generalize these processes by introducing an example of non-Gaussian diffusion in which four mechanisms leading to anomalous diffusion are superimposed that we call 'space-diffusivity-time fractional diffusion'.…”

Local accumulation time for the formation of morphogen gradients from a Lévy diffusion process

“…For this derivative in other particular cases, the authors in [30,31] gave some detailed interpretations. Next, we assume that the waiting-time PDF w(t ) has a finite mean.…”

“…To investigate the dispersal process of Lévy flights of morphogens, we model the jump PDF λ(x) by a Lévy strictly stable density function, which has an infinite jump length variance. The corresponding characteristic function has the form [30,31]…”

“…where η β θ (k) = |k| β e i(sign(k))θπ/2 [30][31][32]. The parameter β (0 < β 2) is an index that describes the stability of a Lévy distribution.…”

“…Furthermore, according to the function η β θ (k), we define a pseudo-differential operator with respect to x: for a sufficiently well-behaved function ϕ(x), define the Riesz-Feller spacefractional derivative [30,34] as…”