Umbral nature of the Poisson random variables

Abstract

The symbolic method, nowadays known as umbral calculus, has been extensively used since the nineteenth century although the mathematical community was sceptical of it, perhaps because of its lack of foundation. This method was fully developed by Rev. John Blissard in a series of papers beginning in 1861 [6–16]; nevertheless it is impossible to attribute the credit for the original idea to him alone since Blissard’s calculus has a mathematical source in symbolic differentiation. In [22] Lucas even claimed that the umbral calculus has its historical roots in the writing of Leibniz for the successive derivatives of a product with two or more factors; moreover Lucas held that this symbolic method had been developed subsequently by Laplace, by Vandermonde, by Herschel, and augmented by the works of Cayley and of Sylvester in the theory of forms. Lucas’ papers attracted considerable attention and the predominant contribution of Blissard to this method was kept in the background. Bell reviewed the whole subject in several papers, restoring the purport of the Blissard’s idea [4], and in 1940 he tried to give a rigorous foundation for the mystery at the bottom of the umbral calculus [5] but his attempt did not gain a hold. Indeed, in the first modern textbook of combinatorics [24], Riordan largely employed this symbolic method without giving any formal justification. It was Gian-Carlo Rota [26] who six years later disclosed the “umbral magic art” consisting in lowering and raising exponents bringing to the light the underlying linear functional. The ideas from [26] led Rota and his collaborators to conceive a beautiful theory which gave rise to a large variety of applications ([23, 27]). Some years later, Roman and Rota [25] gave rigorous form to the umbral tricks in the setting of Hopf algebras. On the other hand, as Rota himself has written in [28]: “…Although the notation of Hopf algebra satisfied the most ardent advocate of spic-and-span rigor, the translation of “classical”, umbral calculus into the newly found rigorous language made the method altogether unwieldy and unmanageable. Not only was the eerie feeling of witchcraft lost in the translation, but, after such a translation, the use of calculus to simplify computation and sharpen our intuition was lost by the wayside…” Thus in 1994 Rota and Taylor [28] started a rigorous and simple presentation of the umbral calculus in the spirit of the founders. The present article takes this last point of view.