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In this work we present an operator $D_\mu$ constructed with the help of thecyclic group set of the $r^{{\small th}}$ roots of unity. This operatorconstitute 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 Besselfunctions of vector index early deeply studied by Klyuchantsev. This paper isargued by specific examples and contains some interesting results which are theprelude of harmonic analysis related to this operator.

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