3 research outputs found
Simply Generated Trees, B-series and Wigner Processes
We consider simply generated trees and study multiplicative functions on
rooted plane trees. We show that the associated generating functions satisfy
differential equations or difference equations. Our approach considers B-series
from Butcher's theory, the generating functions are seen as generalized
Runge-Kutta methodsComment: 19 pages, 1 figur
The normal distribution is -infinitely divisible
We prove that the classical normal distribution is infinitely divisible with
respect to the free additive convolution. We study the Voiculescu transform
first by giving a survey of its combinatorial implications and then
analytically, including a proof of free infinite divisibility. In fact we prove
that a subfamily Askey-Wimp-Kerov distributions are freely infinitely
divisible, of which the normal distribution is a special case. At the time of
this writing this is only the third example known to us of a nontrivial
distribution that is infinitely divisible with respect to both classical and
free convolution, the others being the Cauchy distribution and the free
1/2-stable distribution.Comment: AMS LaTeX, 29 pages, using tikz and 3 eps figures; new proof
including infinite divisibility of certain Askey-Wilson-Kerov distibution