Euler polynomials, Bernoulli polynomials, and Levy's stochastic area formula

In 1951, P. Levy represented the Euler and Bernoulli numbers in terms of the moments of Levy's stochastic area. Recently the authors extended his result to the case of Eulerian polynomials of types A and B. In this paper, we continue to apply the same method to the Euler and Bernoulli polynomials, and will express these polynomials with the use of Levy's stochastic area. Moreover, a natural problem, arising from such representations, to calculate the expectations of polynomials of the stochastic area and the norm of the Brownian motion will be solved. (C) 2011 Elsevier Masson SAS. All rights reserved.