TY - JOUR
T1 - Euler polynomials, Bernoulli polynomials, and Lévy's stochastic area formula
AU - Ikeda, Nobuyuki
AU - Taniguchi, Setsuo
PY - 2011/9
Y1 - 2011/9
N2 - In 1951, P. Lévy represented the Euler and Bernoulli numbers in terms of the moments of Lévy'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 Lévy'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.
AB - In 1951, P. Lévy represented the Euler and Bernoulli numbers in terms of the moments of Lévy'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 Lévy'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.
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U2 - 10.1016/j.bulsci.2011.07.009
DO - 10.1016/j.bulsci.2011.07.009
M3 - Article
AN - SCOPUS:80053320999
VL - 135
SP - 684
EP - 694
JO - Bulletin des Sciences Mathematiques
JF - Bulletin des Sciences Mathematiques
SN - 0007-4497
IS - 6-7
ER -