Fractional integration operator on some radial rays and intertwining for the Dunkl operator

Abstract

Abstract In this paper we consider the differential-difference reflection operator associated with a finite cyclic group, $$Y_\nu f(x)=\frac{df(x)}{dx}+\sum_{i=1}^{m-1}\frac{m\nu_i+m-i}{x}\sum_{j=0}^{m-1}\varepsilon^{-ij}f(\varepsilon^jx).$$ Y ν f ( x ) = d f ( x ) d x + ∑ i = 1 m − 1 m ν i + m − i x ∑ j = 0 m − 1 ε − i j f ( ε 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 Yν 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].