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Hetmaniok E., Pleszczyński M., Słota D., Wituła R.

Journal of Applied Mathematics and Computational Mechanics

2013

Vol. 12, nr 3

93--104

The aim of this paper is to introduce and compare some fundamental analytical properties of the title polynomials. Many similarities between them are emphasized in the paper. Moreover, the authors present many isolated results, new proofs and identities.

Moments of Poisson stochastic integrals with random integrands

Privault N.

Probability and Mathematical Statistics

2012

Vol. 32, Fasc. 2

227--239

We show that the moment of order n of the Poisson stochastic integral of a random process (ux)x∈X over a metric space X is given by the non-linear Mecke identity [formula]. This formula recovers known results in case (u(x))x∈X is a deterministic function on X.

Charlier and edgeworth expansions for distributions and densities in terms of bell polynomials

Withers C. S., Nadarajah S.

Probability and Mathematical Statistics

2009

Vol. 29, Fasc. 2

271--280

We show that the coefficients of the Charlier differential series for distributions and densities are simply Bell polynomials in the cumulants. The same is true for the Edgeworth expansions of distributions and densities of sample means. We use this to obtain higher order extensions of these well-known series.