$r$-extension of Dunkl operator in one variable and Bessel functions of vector index

In this work we present an operator $D_\mu$ constructed with the help of the cyclic group set of the $r^{{\small th}}$ roots of unity. This operator constitute an $r$-extension of the Dunkl operator in one variable because when $r=2$ it reduces to the classical one and admits as eigenfunctions the Bessel functions of vector index early deeply studied by Klyuchantsev. This paper is argued by specific examples and contains some interesting results which are the prelude of harmonic analysis related to this operator.