SÃ©minaire du 22 novembre 2010, 14h00: Katarzyna Gorska,
Université Pierre et Marie Curie.
Exact and explicit forms and combinatorial content of Levy
stable distributions
We briefly recall the origin, history and some physical applications
of Levy
stable probability distributions. We report on recent findings of
exact and
explicit expressions for one-sided (0 < \alpha < 1) and two-sided (1 <
\alpha
\leq 2) Levy stable densities g_{\alpha}(x), of index \alpha for all
\alpha
=
l/k, with k and l positive integers. We shall exemplify analytically
and
graphically several examples of known and infinite ensemble of new
formulae
for such distributions. We observe that in one-sided case (0 < l/k <
1)
g_{l/k}(x) is a solution of the Stieltjes moment problem with
negative
moments being integer combinatorial sequences of factorial
type. This
last property, when seen as a conventional Stieltjes moment problem,
can
be
solved with the use of inverse Mellin transform. In this way we derive
an
explicit formulae for g_{l/k}(x) in terms of Meijer G functions. The
problem
of non-uniqueness of so obtained solutions is discussed. Work done
with
Karol
A. Penson : Phys. Rev. Lett, 2010, in press,
http://arxiv.org/abs/1007.0193
Virginie Collette
Last modified: Thu Nov 18 19:20:44 CET 2010