Umbral nature of the Poisson random variables

Abstract

Abstract. Extending the rigorous presentation of the "classical umbral calculus" [28], the so-called partition polynomials are interpreted with the aim to point out the umbral nature of the Poisson random variables. Among the new umbrae introduced, the main tool is the partition umbra that leads also to a simple expression of the functional composition of the exponential power series. Moreover a new short proof of the Lagrange inversion formula is given.

Symbolic computation of moments of sampling distributions

“…The reader interested in proofs of identities involving auxiliary umbrae is referred to [7]. A feature of the classical umbral calculus is the construction of new auxiliary umbrae by suitable symbolic substitutions.…”

Compactly Supported Wavelets Through the Classical Umbral Calculus

“…In the following, we resume terminology, notations and some basic definitions of classical umbral calculus, as it has been introduced by Rota and Taylor in [12] and further developed in [4]. Fundamental is the idea of associating a sequence of numbers 1, a 1 , a 2 , .…”

“…(d) an element ∈ A, called augmentation such that E[ n ] = δ 0,n , for any nonnegative integer n, where δ i,j = 1 if i = j and 0 otherwise; (e) an element u ∈ A, called unity umbra [4], such that E[u n ] = 1, for any nonnegative integer n.…”

“…We denote the inverse of the umbra α by −1.α , with α ≡ α . For other definitions and properties of umbrae we refer to [13] and [4]. i.e., f (t) e αt , assuming that we naturally extend E to be linear (observe that with this approach we disregard of questions of whether any of the series converge).…”

“…In addition, this new setting led quickly to a nimble language for the theory of random variables (r.v. 's) as showed in [15,4] and in wavelet theory [13].…”