Fractional Calculus, Interpolation, and Traveling Waves: A note (pdf file) on the fractional integro-derivative (FID) regarded as an interpolation of the integral derivatives of the function acted upon–to be more precise, a sinc function interpolation, as in the Whittaker-Shannon interpolation formula, of the integral derivatives, properly normalized. Interference of traveling waves is used to model the action of the FID.

A discussion of the history of the fractional / operational calculus can be found in The Theory of Linear Operators From the Standpoint of Differential Equations of Infinite Order by Harold T. Davis (Principia Press, 1936). Also see F. Mainardi and G. Pagnini’s notes on The Role of Salvatore Pincherle in the Development of Fractional Calculus.

For some connections between fractional calculus, the exponential / Bell / Touchard polynomials (i.e., the Stirling numbers of the second kind), umbral Euler integrals, and confluent hypergeometric functions see the exercises in my notes in my entry below A Generalized Dobinski Relation and the Confluent Hypergeometric Fcts. and also the following MathOverflow and MathStackExchange questions and answers:

2) Pochhammer symbol of a differential, and hypergeometric polynomials