Some remarks on stable densities and operators of fractional differentiation

in Representation theory, dynamical systems, and asymptotic combinatorics, eds/ V. Kaimanovich, A. Lodkin, 117-137, Amer. Math. Soc. Transl. Ser. 2, 217, Amer. Math. Soc., Providence, RI, 2006 Let $D(s)$ be a fractional derivation of order $s$. For a real $p\ne 0$, we construct an integral operator $A(p)$ in an appropriate functional space such that $A(p) D(s) A(p)^{-1}=D(p s)$ for all $s$. The kernel of the operator $A(p)$ is expressed in terms of a function similar to the stable densities.