FRACTIONAL INTEGRATION OPERATOR ON SOME RADIAL RAYS AND INTERTWINING FOR THE DUNKL OPERATOR

In this paper we consider the differential-difference reflection operator associated with a finite cyclic group, Y(v)f(x) = df(x)/dx + Sigma(m-1)(i-1)mv(i) + m - i/x Sigma(m-1)(j-1)epsilon(-ij) f(epsilon(j)x). First we show that the Dimovski ([5], [6]) hyper-Bessel differential operator of arbitrary integer order m is close in frame of the algebra similar to U(sl(2; C)). Secondly, we introduce a difference-differential operator associated to finite cyclic group in the rank one case, and then by using a Poisson-type integral transform proposed by Dimovski and Kiryakova ([7], [11]), we construct a new explicit intertwining (transmutation) operator between the operator Y-v and the derivative operator d/dx. It is to emphasize that both hyper-Bessel operators and the so-called Poisson-Dimovski transformation (transmutation) are typical examples of the operators of generalized fractional calculus [11, 12].