Mediterr. J. Math.
Springer International Publishing AG 2017
The Hypergroup Associated with the
Cherednik Operators and Applications
to the Inﬁnitely Divisible Probabilities
and Convolution Semigroups
Abstract. In this paper we deﬁne the hypergroup (R
with the Cherednik operators on R
and we use the harmonic analysis
on this hypergroup to study the inﬁnite divisible probabilities and the
convolution semigroups on the hypergroup (R
Mathematics Subject Classiﬁcation. 33C67, 51F15, 33E30, 43A62, 43A32.
Keywords. Cherednik operators, Hypergroup, Levy’s continuity
theorem, Inﬁnitely divisible probabilities, Convolution semigroups.
The notion of an abstract algebraic hypergroup has its origins in the stud-
ies of Marty and Wail in the 1930s, and harmonic analysis on hypergroups
dates back to Delsart’s and Levitan’s work during 1930 and 1940, but the
substantial development had to wait till 1970 when Dunkl , Spectar and
Jewett  put hypergroups in the right setting for harmonic analysis. There
have been many fruitful developments in the theory of hypergroups and their
applications in analysis, probability theory and approximation theory (see
In  Cherednik introduced a family of diﬀerential-diﬀerence opera-
tors that nowadays bear his name. We deﬁne in this paper the hypergroup
) associated with the Cherednik operators for a root system and a
multiplicity function satisfying some conditions.
By using the hypergeometric transform and the hypergeometric con-
volution product on the spaces of functions and measures on R
, we deﬁne
the hypergroup (R
) associated with the Cherednik operators on R
study the harmonic analysis on this hypergroup and we prove Levy’s conti-
nuity theorem for the probability measures on R